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BOUGHT WITH THE INCOME OF THE
SAGE ENDOWMENT FUND
THE GIFT OF
HENRY W. SAGE
1891
Mathematics*
Cornell University Library
QA 482.G16 1910
The modern geometry of the triangle.
3 1924 001 522 782
The original of this book is in
the Cornell University Library.
There are no known copyright restrictions in
the United States on the use of the text.
http://www.archive.org/details/cu31924001522782
THE MODERN GEOMETRY
OF THE TRIANGLE.
BY
WILLIAM GALLATLY, MA.
SECOND EDITION.
/<?ro
London :
FRANCIS HODGSON, 89 Faebingdon Street, E.G.
•S
a.
Ms^-rm
PREFACE.
In this little treatise on the Geometry of the Triangle are
presented some of the more important researches on the subject
which have been undertaken during the last thirty years. The
author ventures to express not merely his hope, but his con-
fident expectation, that these novel and interesting theorems —
some British, but the greater part derived from French and
German sources — will widen the outlook of our mathematical
instructors and lend new vigour to their teaching.
The book includes some articles contributed by the present
writer to the Educational Times Reprint, to whose editor he
would offer his sincere thanks for the great encouragement
which he has derived from such recognition. He is also
most grateful to Sir George Greenhill, Prof. A. C. Dixon,
Mr. V. R. Aiyar, Mr. W. F. Beard, Mr. R. F. Davis, and
Mr. E. P. Rouse for permission to use the theorems due to
them.
W. G.
SYNOPSIS OF CHAPTERS.
[The numbers refer to Sections.]
CHAPTEB I. — DlBECTION ANGLES. (1-13.)
Relations between 6 X 6 3 9 3 and pqr : lines at rt. angles, condition: re-
lation between pqr : distance between (0^7) and (a'ff'y') : perpr. on line :
Properties of Quad. . Centre Circle.
Chapteb II.— Medial and Tbipolab Coobdinates. (14-21.)
Medial formulas : Peuerbach Point : mid-point line of quad. :
N.P. Circle : Tripolar eqn. to circumdiameter : straight line (general) :
circle : to find points with given trip. c. : trip. c. of Limiting Points.
Chapter III. — Pobistic Triangles. (22-34.)
Poristic condition : R.A. of 0(R) and I(r) : points S) S^trHiGi poristically
fixed : circular loci of FGHOTNM : properties of Nagel Point N :
Gergonne Point M : poristic formulas : locus of M found.
Chapter IV. — Simson Lines. (35-55.)
Draw S.L. in given direction : S.L. bisects TH: OAT = a x : perps. from
A, B, C on S.L. : eqn. to S.L. : perpr. TV, length and direction : the
point Niov quad. : pairs of S.L. . the point a : circle (kkf) : eqn. to TOT' ;
ABC and A'B'C n.c. of w : the centre Oi of (kh') : tsicusp Hypocycloid :
Cubic in sin</> or cos* : Greenhill and Dixon's Theorem.
Chapteb V.— Pedal Triangles. (56-69.)
Pedal triangle def similar to LMN : B8C = A + k : sin A a art : Limiting
Points : n in terms of xyz : Radical Axes (group) : Feuerbach's Theorem :
V, n, OS 2 , and (a&y) in terms of A^c : Artzt's Parabola : So for S lt
inverse of S : S.L. of T, T' are axes of similitude for def, d 1 e 1 f i .
VI MODERN GEOMETRY.
Chapter VI. — The Obthopole. (70-80.)
Concurrence of pp', qq', rr' : S = 2B cos 9, cos » 2 cos e 3 ; S found geo-
metrically : general n.c. of S : point <r lies on N.P. circle : ABC and
A'B'C n.c. of a : a coincides with a : three S.L. through a point,
reciprocal relations : Lemoyne's Theorem : pedal circles through a (as) :
constant b.c. of S for def: Harmonic system of lines, a common to the
4 circles. (See Appendix I and II.)
Chapter VII. — Antipedal Triangles. (81-87.)
Angular c. : Orthologic and Antipedal triangles : no. of S' : V = \M :
n.c. of centre of similitude of U and V : similar properties of Si and S\ :
Vi— V\ = 4a ; 8' and S\ are called Twin Points.
Chapter VIII. — Orthogonal Projection of a Triangle. 88-100.)
Shape and size of projection of ABC on plane passipg through fixed axis
in plane of ABC : ABC projected into triangle with given angles : projec-
tion of ABC on planes at constant inclination : 2o' 2 cot A = 2 A (1 + cos 2 «)...:
equilat. triangle and Brocard Angle : Antipedal triangles and projection :
calculation of a', b', c' : Pedal triangles and projection : general theory
for any triangle XYZ : Sehoute Circles.
Chapter IX. — Counter Points. (101-116.)
a', II', D', M', q' in terms of \/j.v : nn' = 4E 2 g 2 : equation to minor
axis: SA. S'A = AB.Ac : A'B'C b.c. of cr : p- = Ja/JV : Ratio Bl : CI :
pedal circle SS' cuts N.P. circle at a,u : Aiyar's Theorem, OS.OS' = 2iJ.OV :
M'Cay's Cubic : Counterpoint conies, direction of asymptotes and axes :
For TOT, conic is R.H. ; centre, asymptotes, (semi-axis) 2 : similarly for
S' and S'i : join of Twin Points S] S\ bisected at a ; circle centre S
radius SS\ passes through I'm'n'.
Chapter X.— Lemoine Geometry. (117-129.)
K found : n.c. and b.c. of K: E centroid of def: equation to OK: n :
a 2 + j8 2 + 7 2 and m 2 + 2 + «> 2 each a min. at if: Artzt's Parabola: triangle
TiT s T 3 : Lemoine Point of IiI 2 I 3 : list of " {s-a) " points : AK bisects
chords parallel to T 2 T 3 : Harmonic quad. . A'l passes through K : locus
of centres of rect. inscribed in ABC : cos0! a a(6 2 — c' 2 ) : tripolar equation
to OK ; Apollonian Circles : Lemoine Axis : Harmonic quad, inverts into
square.
SYNOPSIS OF CHAPTERS. Vll
Chapter XI. — Lemoine-Brocard Geometry. (130-152.)
Forms for cot a : n.o. and b.c. of n and n' : equation to an' : a not
greater than 30° : useful formulse : Neuberg Circles : Steiner Angles :
Pedal triangle of a : Triangles XBC, YCA, ZAB (see also Appendix III) :
First Brooard Triangle PQB : centre of perspective D : eqn. to axis of
perspective : G double point of ABC, PQB : OK bisects nn' at right
angles : On = eB, OK = eB sec a> : Steiner and Tarry Points : figure
KPBOQ similar to 2ACTB: O mean centre of points yzx, nxy, xyz:
apply to PQB ; G oentroid of Dan' : D lies on 2 OT : OD = e-B :
Isodynamic Points S and 5, : Isogonic points 3' and S,' : Circum-Ellipse
and Steiner Ellipse.
Chapter XII. — Pivot Points. Tucker Circles. (153-166.)
Pairs of homothetic triangles, inscribed and circumscribed to ABC :
family of circles touch conic : Tucker Circles : list of formulse : Radical
Axis : First Lemoine Circle : Pedal circle of aa' : Second Lemoine Circle :
Taylor Circle : trip. c. of Limiting Points for Taylor Circle are as
cot A, cotS, cot C.
CHAPTER I.
INTRODUCTION : DIRECTION" ANGLES.
1. In this work the following conventions are observed: — the
Circumoentre of ABG, the triangle of reference, will be denoted
by ; the Orthocentre by M; the feet of the perpendiculars
from A, B, G on BG, CA, AB respectively by H it H 2 , H s (the
triangle ffjffjfl, being called the Orthocentric Triangle) ; the
lengths of AM V BH„, CH S by h lt h % , h 3 ; the Centroid or Centre
of Gravity by G ; the in- and ex-centres by I, J,, J 2 , J s ; the
points of contact of the circle I with the sides of ABG by X, Y,
Z ; the corresponding points for the circle I, being X„ x„ Z\.
The Circumcentre of I-J^ i s «7j 'which lies on 01; also
OJ = 01, and the Circumradius of J^Jg = 2B.
B
2 MODERN GEOMETRY.
The mid-points of BO, GA, AB are A', B', C ; the triangle
A'B'C being called the Medial Triangle. Its Circumcircle is
the Nine-Point Circle, with Centre 0' and radius = \R ; the
Orthocentre of A'B'C is 0, the Centroid is G, while the in-centre
is denoted by 1'.
The lines drawn through A, B, parallel to BO, OA, AB
form the Anti-Medial Triangle A-fi-fix.
Its Circumcentre is H, its Centroid is G, its Medial or Nine-
Point Circle is ABO ; and its Nine-Point Centre is 0.
The letters " n.c." stand for normal or trilinear coordinates
denoted by a/3y.
The letters " b.c." stand for barycentric or areal or triangular
coordinates, denoted by xyz ; also x = aa, &c, so that
x + y+z = 2X
2. Let p, q, r be the lengths of the perpendiculars from A,B,
on a straight line 2'2"
Let $x, # 2 , a be the Direction Angles of TT' ; that is, the
angles which the sides of ABO make with TT', these angles
being measured from TT' as axis, and in the same sense.
By projecting the sides of ABO along and perpendicular to
TT', we obtain 2 . a cos 0, = 0, 2.asin(9 1 =0.
The diagram shows that a sin 6 X = q— r, &c, so that
2 . ap sin &x= %.p (q—r) = 0.
Two sets of direction angles should be particularly noted.
For 01, cos Ox = A'X/OI = \ (b-c)/0I oc (6-c).
For OGH,
cos 6x = A'HxfOH = li ain(B-C)/0H cc (tf-c^/a.
DIRECTION ANGLES. 3
To express cos 6 X in terms of p, q, r.
Since Off, : H X B = b cos . c cos B,
.■. q.b cos G+r.c cos B = (& cos 0+c cos P)!?^
= a(p — H 1 d') = ap— a.AH l cos6 l ;
.'. 2A.cos 6, = ap— 6g cos — or cos B;
When ff, falls outside dd', the right-hand signs are changed.
3. To determine the condition that la.+vi/3 + ny = 0, and
Va+m'P+n'y = may be at right angles.
Let 6 U 6 2 , 8 S and <£„ <£ 2 , <£ 3 be the direction angles of the two
lines, so that 8 1 = 4>i ± \*-
Let p, q, r, p', q', r' be the perpendiculars on the lines from
A, B, G, so that Zoc ap, V oc ap'.
Now op' sin 0J+... =p'(q' — r') + ... = 0.
And 2 A . sin <£, = 2A . sin (8 1 ±%n) = 2A cos ^
= ap — bq cos G — or cos P.
.'. ap' {ap—bq cos C—cr cosP) + . .. = 0,
or ZZ' + mm'-f«ra' — (tim' + mV) cos.4 — ... =0,
which is the required condition.
4. To determine it, the length of the perpendicular on TT' from
a point P, whose b.c. are (x, y, z), in terms of p, q, r.
Since P is the centre for masses at A, B, G proportional to
^■>y> z - /. (x + y + z)-rr=px + qy + rz;
so that 7r is determined when p, q, r are known.
Note that the ratios only of x, y, z are needed.
5. To determine it, when TT' is la + mfi+ny = 0.
Put Z 2 +...— 2mm cos 4— ... = D 2 .
Now I a: ap = k.ap.
Also 2 (a°p 2 — qr.2bc cos A) = 4A 2 ,
and x + y + z = aa+&/3+cy = 2A.
Hence * ir = (Za+m/3 + ray)/D.
A form of little use, as it is almost always difficult to
evaluate B.
4 MODERN GEOMETRY.
6. A straight line TT' is determined when any two of the three
perpendiculars p, q, r are given absolutely. It follows that
there must be some independent relation between them.
From elementary Cartesian Geometry we have
2A = Ap.qr + Bq.rp+Cr.pq
= p.acos$ 1 + q.b cos 8 2 + r.c cos 3
.-. 4A 2 = op.2A cos #!+... = ap (ap—bq cos G—cr cos B) + ...
= ^ (ay -2beooaA.gr) = 2{aV-(-a 2 + fc 2 + c 2 )gr}.
This is the relation sought.
When TT' passes through A,p = 0,so that
Z,y + c y — 26c cos A . qr = 4A 2 .
7. The points P, P ' lying on TT' have absolute n.c. (a, /?, y ) and
(a/ /3', •/)• It is required to determine d, the length of PP', in
terms of these coordinates.
We have ^{ay-i-a' + W + c^qr} = 4A 2 .
But a 2 = 2 A (cot P + cot 0),
and -a 2 +6 2 + c 2 = 4A cot A
Hence (q—rY cot A+ (r—pY cot B + (p—qY cob G = 2A.
Now q — r = a sin 0, = a. (a — a')/d.
Hence
A/E 2 . d' 2 = (a-a'Y sin 2A + (B-B'Y sin 2B + (y — y'Y sin 2(7.
8. To prove that, when TT' is a circumdiameter,
ap/B = 6 cos 3 +c cos 3 (Gr)t
From (2) we can express the right side in terms of p, q, r.
Then apply the condition a cos A .p+... = 0, and the result
follows.
Hence prove that —
for 01, p = B/0I.(b-c)(s-a)/a ;
for OH, p = B/OH. (6 2 -c 2 ) cos A/a.
The equation to TOT', which is p.aa + ... = 0, now takes
the form (6 cos# s +c cos 2 )a+ ... = 0.
For 01, (6-c)(«-a).a+... =
[more useful that (cos P — cos G) a+ ... = 0] .
For OGR, (& 2 -c 2 )cos4.a+... = 0.
I For theorems and proofs marked (G) the present writer is responsible.
DIRECTION ANGLES.
9. The results just obtained are useful in investigating some
algebraic relations of a quadrilateral, which has BG, OA, AB
for three of its sides : the fourth side beiDg PQJR.
The circumcircles of the four triangles AQB, BBP, GPQ,
ABO, have a common point (call it M) so that the parabola
which touches the four sides of the quadrilateral will have M
for focus ; while the four orthocentres lie on the directrix, which
is also the common radical axis of the circles (AP)(BQ)(OB).
The four circumcentres 0, O u 2 , 3 are known to lie on a
circle (call it the Centre Circle) which also passes through M.
ft
Let p be the radius of this circle and p v Pi , p a the radii of
AQB, BBP, GPQ.
Since the angles at Q, B are # 2 , S ,
'2p 1 sin S = AQ (in circle AQB) — p/ain 2 ;
.-. p 1 = p sin 0j . 1/(2 sin 6 X sin 2 sin # a ) = m.p sin 6 V
In circle AMQB, AM = 2 Pl sin ABM,
and in BMBP, BM = 2 Pi sin (BBM or ABM) ;
.-. AM : BM : CM = Pl : /> 2 : p 3 = p sin 6 1 : q sin 2 : r sin 8 ;
therefore the n.c. of M are as
1/p sin 6 V 1/q sin 2 , 1/r sin 3 ,
or a/p(q—r), Sue. or a/(l/q—l/r). (R.F.Davis)
Note that in the circle AMQB,
QB = 2 Pl sin A <x ap sin V
6 MODERN GEOMETRY.
10. The join 00, of the centres ABGM, ABQM is perpendi-
cular to the common chord MA.
Similarly 00, is perpendicular to the chord MB.
.-. aO,00, = AMB-AGB (in ABGM) = G ;
.-. 0,0,0, = G, ... ;
and the triangle 0,0,0, is similar to ABC. But, from
MA : MB : MG = Pl : P , : p s = MO, : MO a : MO,.
Hence M is the double point of the similar triangles ABO,
0,0,0,.
Since 0,0 is perpendicular to MB, and 0,0, to MB,
.-. / 00,0, or 0M0, = BMB = BPB (circle BMPB) = 6 V
Hence the chords 00,, 00„ 00, in the Centre Circle subtend
angles 6„ 0„ 0„ so that
00, : 00, : 00, — sin^ : sin0 2 : sin 6,.
11. To determine the equation of the 4-orthocentre line. The
equation to PQB being p.aa + q.bB+r.cy = 0, the point P is
determined by u. = 0. q.bB+r.cy = 0.
The perpendicular from P on AB proves to be
cos A . aa/r + (c/p — a cos B/r) B + cos A . cy/p = 0,
and the perpendicular from B on AG is
cos A . aa/q + cos A . bB/p + (b/p — a cos C/q) y = 0.
By subtraction their point of intersection is found to lie on
(1/q—l/r) oosA.a+(l/r-l/p) cos B .B+ (1/p — l/q) cos C.y,
or p gind, cos A.aa.+ ... = 0,
or cos^4/a'.aa+ ... = 0,
or p(q— r) cos A. a+... = 0. (M is a'B'y')
From the symmetry of this equation the line clearly passes
through the other three orthocentres and is therefore the
directrix of the parabola.
For example, let PQR be x/a?+... = 0, which will prove to
be the Lemoine Axis (128).*
Here p or 1/a 2 • so that the focus of the parabola, known as
Kiepert's Parabola, has n.c. a/(6 2 — c 2 ) &c, the directrix being
(6 2 -c 2 )cos^L.a+... =0, which is OGH.
The mid-points of diagonals lie on
( — l/p + l/q + l/r)x+...=0, or cot A.x+... = 0,
the well known Badical Axis of the circles ABG, A'B'G, &c.
* The bracketed numbers refer to sections.
DIRECTION ANGLES. 7
12. To determine p, the radius of the Centre circle. (G)
M0 1 = p, = m.p sin 0„ so M0 2 =■ m.q sin 6 2 ,
and 0,-MO,, = 0,0a = 0;
.-. OiOJ/m? = p 2 sin 2 1 + 2 2 sin !! 2 -2pgsin(9,sin0 2 cos(7
_ p\q-rf qHr-pf _ J q - r )(p-r) a 2 + 6 2 -c 2 .
a 3 " r 6 2 P? ah ' ab '
.-. 0,0^/mV = [2 a i q*r>-$p i qr (-a' + b' + c')]/* 1 = tf/aW.
Bat since Z 0,0 3 2 = 0, Ofi^ = 2 P sin 0,
.". p = mR.h 3 /abc.
To determine the length of AP, one of the diagonals of the
quadrilateral.
The distance d between (afty) and (a'fty') is given by
d\ A/fl 2 = (a-a') 2 sin 24 + ... • (6)
Now, for P, u = 0; q.bp + c.ry = 0;
■•■ )8=(-r)/(g-r).2A/6, y = g/( 9 -r) .2A/c.
Also for 4, a' = 2A/a, ft = 0, y' = ;
.-. 4P' 2 .A/E 2 = 4A7a a .sin2^+rV( 3 -r) 2 .4A76 2 .sin2B
+ 97(g-r) 2 .4A 2 /c 2 .sin2C.
Hence AP* (q—r) 2 — qV + r l c i —2qrbc cos A.
13. Let coj, w 2 , o) s be the centres of the diameter circles, or
mid-points of AP, BQ, CB. To determine the length of a>,(o 2 .
Foreo,, a 1 = A/a, ft = (-r)/(g-r). A/6, y, = g/( ? -r).A/c.
Foro) 2 , a 2 = r/(r-p).A/a, ft = A/6, y 2 = (_-p)/(r—p). A/c.
Now, ^V . A/E 2 = K-a,) 3 sin 24 + . . .
/j 2 A 2 _. „ , , <r 2 A 2
•^^T^T.^-™ 2 *
(_p-r) 2 a 2 "" (g-r) 2 6'
r 2 (p— a) 2 A" . or)
(js — ry(q — r)- <?
.: 2 . ui.V/A = , - P " , . cot 4 + r-2-— . . cot £
+ . ^^-?>\, .oot0.
Hence, since cot 4 = ( - a 2 + 6 2 + c 2 )/4A,
8.^{q-ry(r-py = p 2 (g_r) 2 (-a 2 + 6 2 + c 2 ) + ... ;
.-, 4.<o,<o 2 2 ( ? -r) 2 (r-p) 2 = 2a 3 2 V 2 -2^y (-a 3 +6 2 +c 3 ) = &;
.: 10,(0, = ^/[fa-O^-p)] «.p-g cc csin0 8 ;
(o 2 (o s : (o s (o, : odjWj = a sin #, : 6 sin 2 : c sin a .
CHAPTER II.
MEDIAL AND TRIPOLAR COORDINATES.
14. Medial Coordinates. — If A', B', C are the mid-points of
BG, GA, AB, the triangle A'B'G' is called the Medial Triangle
of ABO ; its circumcircle A'B'G' is the Nine-Point Circle whose
centre 0' bisects OH.
Eor every point P in ABO there is a homologous point P'
in A'B'G', such that P' lies on OP, and GP = 2. OP'.
Let aBy, a'B'y' be the n.c. of a point P referred to ABG, A'B'G'
respectively.
A diagram shows that a+a' = ±h v
.■. aa + aa' — \.ah r = A = 4. area of A'B'G'
= 2(a'a' + b'B' + c'y') [a' = ia]
= aa' + 60' + cy';
.-. aa = bB' + cy', &c,
so that 2aa' = — aa + bB + cy,
or, in b.c,
x = y'+z', 2x' =— x + y+z.
For example, the A'B'G' b.c. of the Feuerbach Point F being
a/(b—c), b/(c—a), c/(a—b), to determine the ABO b.c. of this
point.
Here x' <x a/(b — c), &c;
.-. x = y' + z 1 oc b/(c-a) + c/(a-b) cc (b-c)(s-a)/(c-a)(a-b)
oc (6_ c )2( s _a).
If the A'B'G' b.c. are to be deduced from the ABG b.c, then
2x' = -x + y + z<x -(6-c) l (e-a) + (c-a) s («-6) + (o-6) , (s-c)
a a(a— b)(a— c) a a/(b—c).
15. If the ABC equation to a straight line is Ix+my+nz = 0,
the A'B'G' equation is
Z(y +«') + — = 0, or (m+») »'+... = 0.
If the ^'.B'C equation to a straight line is I'x' + mtfy'+n'z' = 0,
the A HO equation is
l'(— x + y + z) + ... = 0, or ( — V + m' + ri)x + ... = 0.
8
MEDIAL AND TKIPOLAR COORDINATES. 9
Example. — The well known Radical Axis whose ABC equation
is cotA.x + ... — becomes
(cot B + cot C)x' + . . . = or aV + b 2 y' + c*z' = 0,
when referred to A'B'C.
If H. v Hj, H s are the feet of perpendiculars, the ABG
equation to H^H S is
— cot A .x + cotB.y + cot G.z = 0.
Therefore the A'B'C' equation is
(cot B -\-cot C)x+ (cot G— cot A)y + (—cot A + cot B)z = 0,
reducing to a"x+ (a 2 — c 2 ) y+ (a 2 — 6 2 ) z.
Returning to the quadrilateral discussed in section (13) we
see the b.c. of o^ given by a ; = A/a, &c.
Hence the ABG equation to the mid-point line of the
quadrilateral is ( — 1/p + 1/q + 1/r) x + . . . = 0.
The A'B'C' equation therefore is
#'/p + 2/'/2 + z '/ r = 0, or aqr.a+ ... = 0.
The perpendicular on this from A' is therefore given by
7Tj = aqr.hJD,
where D 2 = S { a 2 g 2 r 2 -jp V( - a 2 + & 2 + c 2 ) } = fc 6 .
.*. 7r,p = 2A.pqr/h? = ir^ = 7r s r.
16. To determine the A'B'C' equation to a circumdiameter
TOT', whose direction angles are 6 U a , 3 .
Let/, q', r' be the perpendiculars from A', B', C on TOT'.
A diagram shows that
p' = 0.A' cos X = JS cos ^4. cos 6,.
Hence the required equation is
cosAcos$ 1 .x' + ... —0.
Example.— For OI, cos 6 1 = %(b—c)/0I.
Hence the A'B'C' equation to 01 is
(6— c) cosA.x'+ ... = 0.
The ABC equation to the circumcircle ABC is
a/a + ... = 0, or a?/x+... = 0.
Therefore its A'B'C equation is
a?/(y' + z') + ... = 0,
or aV + &V + cV + (a 2 + V + c 2 ) (y V + z'x' + x'y') = 0.
Referred to A'B'C, the equation of the Nine-Point or Medial
Circle is a?/x'+... = 0.
10 MODERN GEOMETRY.
Referred to A BO, this becomes
a?/(~m + y +*) + ... = 0,
reducing to the well known form
a cos A.a? + ... — a/3y— ... = 0.
17. Tripolar Coordinates. —
The tripolar coordinates of a point are its distances, or ratios
of distances, from A, B, 0.
To determine the tripolar equation to a circumdiameter whose
direction angles are V 2 , S .
Let P be any point on the line, x, y, z, the projections of OP
on the sides.
Then, if r„ r 2 , r s be the tripolar coordinates of P,
r a 2 — r s 2 = a. 2x = a. i . OP cos V
and irj-rf)r*+W-r*)r t +\r?-r*)rf = 0;
.•. a cos 1# r! 2 + b cos 6^.r* + c cos a .r s 2 = 0,
and acos0 x +... = 0.
So that an equation of the form
lr* + mr^ + nr s 2 = 0,
where l+m+n = 0,
represents a circumdiameter.
Note that, for every point P or (r„ r„ r 3 ) on the line, the
ratios r^—r^ : r^—r* : r^—r^ are constant ; being, in fact,
. equal to the ratios a cos 1 : b cos 2 : c cos 3 .
The tripolar equations to 01, OGH should be noted,
(i) The projection of 01 on BO — £ (b—c) ;
.'. cos 1 oc (b—c) ;
and the equation to 1 is
a (6— c)?-, 2 ^-... =0.
(ii) The projection of OH on BO
= (6 2 -c 2 )/2a;
and the equation to OS is
{V-t>)r*+...=0.
18. To find the equation to a straight line with direction
angles 6 V 2 , 0„ and at a distance d from 0. (G)
Transferring to Cartesian coordinates, we see that the equa-
tion differs only by a constant from that of the parallel
circumdiameter.
MEDIAL AND TRIPOLAB. COORDINATES.
11
It must therefore be of the form
a cos l .r- 1 2 + ... = k.
Let A', B', C'be the mid-points of the sides, and let^A'O meet
the line in A".
Then if (p„ p 2 , p 3 ) be the coordinates of A",
k = a cos 0, . pf + 6 cos&,.p 2 3 +ecos0 3 .p 2 2 [p 2 = Pa J
= a cos 0j (pj 5 — p 2 2 )
= acos« I (^if 2 -ilf J B 2 )
= a cos 0,. 2c. CM"
= a(OD/0.4")2e.CM"sin7*
= <2 . 2ac sin 7?
= 4eZA.
Hence the required equation is
acos0].rf+... = 4cZA.
19. When (l+rn+n) is not zero.
To prove that if Q be the mean centre of masses Z, m, « at
-d, B, G; or if (Z, m, m) are the b.c. of Q, then, for any point
P whose tripolar coordinates are (r 1( r„ r 8 ),
Zr, 2 + mr ! s +«r, , = Z.4Q s +m.BQ ! +».0Q 2 + (l + m+n) PQ\
Take any rectangular axes at Q, and let (o^a,), (bfi,), (<y t ),
(ccy) be the Cartesian coordinates of 4, 5, G, P.
Then )!»•,• = Z(a 1 -aO a + Z(a,-2/) s
= l.AQ'+l. PQ'-2x.la 1 -2y Ja v
But, since (^ is the mean centre for masses Z, to, w,
•. Zoj + to6, + wCj = ; Za 2 +'mb 2 + wc 2 = q ;
\2 MODERN GEOMETRY.
.-. k'+mrf+ur* = l.AQ> + m. BQ* +n.CQ? + (l + m+n) PQ 2 -
This, being true for any point P, is true for ;
.: l.R i +m.R 2 +n.R 2 = S,l.AQ , ' + (l + m + n)OQ i ;
.: k-* + mr* + nr* = (Z + m + «)(QP 2 -Q0 2 + E 2 ).
The power II of the point Q for the circle ABO is E 2 — OQ 2
or 0Q?—R 2 , according as Q lies within or without the circle.
If P describes a circle of radius p(— PQ) round Q, an internal
point, then the tripolar equation to this circle is
lr* + mr; + nr s 2 = (l + m + n) (p 2 + H).
If the circle cuts the circle ABO orthogonally, then
OQ 2 =E 2 + P 2 ,
so that the circle becomes
fr a 2 + bm-, ! + nr* = 0. (R. F. Davis) .
Examples. — •
(A) The circumcircle :
Here p = R, QO = 0, I <x sin 2A ;
.-. sin24.r 1 2 + sin2B.r s 2 +sin20.r 8 2 t= 4A.
(B) The inscribed circle :
p = r, QO' = 10* = R*-2Rr, I ex a;
.: ar* + br* + cr? = 2A (r + 2E) .
(C) The Nine-Point circle :
P = iR; Q0* = \OB? = JE 2 — 2E cos A cos B cos (7.
Since the n.c. of the Nine-Point centre are cos (B — 0)....,
the b.c. are sin ^. cos (B— G),..., so that
Z <x sin2P + sin20;
.-. 2 (sin 2B+sin 20) r°- = 4A (1 + 2 cos A cos B cos 0).
20. To determine the point or points whose tripolar coordi-
nates are in the given ratios p : a : r.
Divide BO, GA, AB internally at P, Q, R and externally at
P', Qj, E', so that
BP :GP = q:r = BP' : OP',
GQ:AQ = r:p= GQ' : AQ',
AR:BR=p:q = AR': BR'.
Let <o„ o) 2 , <o s be the centres of the circles described on PP',
QQ', RR' as diameters, and let the circles (PP), (QQ) intersect
at T, T'.
MEDIAL AND TR1P0LAE COORDINATES. 13
Then since GPBP' is harmonic, we have, for every point T or
T' on the circle PP'
% a> 3
BT : GT = BP : CP = q:r-BP' : CP' = BT' : CV,
and TP, TP' bisect the angles at T, while T'P, T'P' bisect those
at r.
So for every point T or T on the circle (QQ')
GT: AT = GQ:AQ = r:p, &o.
Hence at T, T' the points of intersection of the circles (PP'),
(QQ) AT:BT:GT-p:q:r- AT' : BT' : CT'.
The symmetry of the result shows that T and T' lie also on
the circle (BR').
Hence there are two points whose tripolar coordinates are as
p : q : r, and these points are common to the three circles
(PP'), (QQ'), (BB').
Since (GPBP) is harmonic,
.-. «,) 1 P ! = (0,5.(0,0.
Hence the circle (PP'), and similarly the circles (QQ'),
(BB'), cut the circle ABG orthogonally, so that the tangents
from to these circles are each equal to B.
It follows that —
(a) lies on TT', the common chord or Radical Axis of
the three circles (PP'), (QQ'), (BB 1 ).
(b) OT. OT' = B 2 , so that T, V are inverse points in circle
ABO.
(c) The circle ABG cuts orthogonally every circle through
T, T including the circle on TT' as diameter, so
that u>T* = Ooi 2 —B}, where <d is the mid-point of
TT'.
14 MODERN GEOMETRY.
(d) The centres a^, cu 2 , w s lie on the line through a>, bisect-
ing TT' at right angles.
21. The tripolar coordinates of Limiting Points. (Gri
Since OT. OF = R\ the circle ABO belongs to the coaxal
system which has T, T' for Limiting Points, and therefore
(UjWjWj for Radical Axis ; so that, if ir v w 2 , w s are the perpen-
diculars from A, B, on m^o^, we have by coaxal theory
2.0T.1T, = AT l or ^ccp*.
Hence the equation to the Radical Axis (OjWjWj is
p'x + q'y+r'z = 0.
And conversely, if the Radical Axis be
\x + fny + vz = 0,
then the tripolar coordinates of T or T' are Vk, v/yu, */v.
Examples. —
(1) For the coaxal system to which the circle ABO and the
in-circle XTZ belong, the Radical Axis is
(s— a,yx+... = 0.
Hence the tripolar coordinates of the limiting points lying on
01 are as (s— a), (s— 6), (s— c).
(2) For the circles ABO, IJ^ the Radical Axis is
a + B + y = or %/a+... = 0;
.-. p : q : r = 1/ Va : 1/ Vb : 1/ Vc,
the limiting points lying on 01.
(3) The circle ABO and the Antimedial circle A 1 B 1 G l (1)
have Radical Axis d?x + Wy + (?z — ;
.•. p : q : r = a : b : c,
the limiting points lying on OOH.
(4) The circles ABO, A'B'C, Polar Circle, &c, have for their
common Radical Axis
cot A .x + cot B .y + cot .z = ;
.-. p : q : r = \/cot^l : VcotB : i/eot 0,
the limiting points lying on 0GH.
CHAPTER III.
POKISTIC TRIANGLES.
22. Let J be any point on the fixed diameter BB' of the circle
0(B). With centre B and radius BI cut the circle at e x and
e 2 ; let e,e a cut BB' at e. Let 01 = d, and Ie = r.
Then De + eI+/0 = E, and Be = De>/2B = DP/2R;
.: (B—df/2B+r+d-B;
.: r = (B*-d*)/2B, or OF = d* = R'-2Br.
(Greenhill)
♦ L'
An infinite number of triangles can be inscribed in the circle
0(B) and described about the circle I(r), provided
01 2 = B 2 -2Br.
On 0(B) take any point A, and draw -tangents AB, AG to
I(r). Let BI, 01 meet 0(B) in B', 0' respectively.
15
16 MODERN GEOMETRY.
Then, since BI.IB' = R?-OP = 2Rr, and BI = r/sinO ;
.-. B'l = 2R sin 6 = B'A.
So O'l = G'A.
Hence B'O' bisects AI at right angles, so that
L B'C A = B'C'I or B'G'G ;
.-. £B'BA= B'BG;
;. 'BO touches I(r).
It follows that by taking a series of points I along DD' and
calculating r from r = (E 2 — d 3 )/2E, we have an infinite
number of circles I(r) ; each of which, combined with 0(R),
gives a poristic system of triangles.
23. The Radical Axis of 0(R) and I(r).
Let L, L' be the Limiting Points of the two circles, and let
EJS V the Radical Axis, cat 01 in E.
Bisect 01 in k.
Then SO 2 - K 2 = EV = EP-r\
by ordinary coaxal theory ;
.-. 2d. Eh = EO^-EV = E 2 -r 2 ;
.-. SO = Efc + ^d = (2E 2 -2flr-r 2 )/2d,
and SI = Sfc-id = (2Rr-r 2 )/2d.
Also Si 2 ■= SI 2 - r 2 = r s (4E + r)/4tf 2 -
24. We now proceed to discuss some points which remain
unchanged in a system of poristically variable triangles ABO.
(a) The inner and outer centres of similitude (#, and S 2 )
of the circles ABG, XYZ.
(b) The centre of similitude (<r) of the homothetic triangles
XYZ and I.IJs-
(c) The orthocentre (Hi) of the triangle XYZ.
(d) The Weill Point ((?;), the centroid of XYZ.
25. (a) To determine the distances of Sj and $ 2 from the
Radical Axis.
Since 01 is divided at S lt so that
0S t : IS l = R:r.
.: ES l (R+r) = EI.R + EO.r;
. .jbd _ f4fl + r)(E-r)r
1 2d(E+r) '
JPOKISTIC TRIANGLES.
17
Similarly
ES<
_ r*(B + r)
2d(E-r) '
So that the circle (5,/Sj) is coaxal with 0(B) and I(r), a well
known theorem.
26. To show that <r, the centre of similitude of the homothetic
triangles XYZ, 1^1^, is poristically fixed, and to determine its
distance from the Radical Axis.
Since the oircumcentre of XYZ is I, while the circumcentre
of I X I 2 I S is J T lying on 01, and such that 0J= 01;
.•. n- also lies on 01;
and aljaj = ratio of circumradii = r/2B — constant ;
.". <r is & fined point.
Again o-I/IJ = r/(2B—r), from above ;
.-. <rl = 2dr/(2B-r) ;
^(AB+r)
E<t-EI—<tI =
2d(2H—r)'
18 MODERN GEOMETRY.
And E.I=r(2R-r)/2d.
.-. E*.EI=EI?, (Greenhill)
so that the circle la- belongs to the coaxal system. Note that
the homothetic triangles XYZ, Jj/jJ, slide on fixed circles, the
joins .XT,, Y7 2 , ZI S passing through the fixed point o\
It will be convenient here to determine the n.c. of the point <r.
From figure p..-, drawing a-n 1 perpendicular to ]i0, and
noting that X and I x are homologous points in XYZ, I^Jj, we
have o-X/o-Ii = ratio of circumradii of the triangles
= r/2B ;
.-. ctctj/JjX, = a-X/I.X = r/(2B-r).
(To-, s a = r/(2B—r) .r, ;
.-. a : /3 : y = rj : r 2 : r 8 = l/(s — a) : 1/0— 6) : l/(s— c).
Note also that, since Zand >7are the circumcentres of XYZ,
.-. al/trj = r/2B ;
.-. b-I/7/ = r/(2E-r), and IJ=2.0I=2d;
'.-. 0-7 = 2dr/(2R-r).
27. To prove that 17;, the orthocentre of XY.Z, is poristically
fixed, and to determine its distance from the Radical Axis.
Since 77; and 7 are the orthocentres of XYZ, J^Z,,
.'. Hi lies on a-I, that is, on 01 ;
a-HJa-I = r/2R, a fixed ratio ;
.•. Z7j is a fixed point.
Again, since a-I = 2dr/(2B— r), (26)
.-. 0-77* = r/2B . o-I = — *! ;
' B(2B-r)'
also E<r = im±Z±. (26 )
2d{2B-r)' K >
EH, = Ev+vHi = **(**+') +
2d(2B—r) R(2B-r)'
= 3S/2d. '
Note also that
77,7 = El -EH = 2Er ~ ra _ *f
2d 2d'
= ri/72.
PORISTIC TRIANGLES.
19
To determine the n.c. of if,.
The orthocentre H t of XYZ is the centre of masses tan X,
tan Y, tan Z placed at X, Y, Z.
;. a oc tanY. X Y sin .Z+tan Z. ZX sin Y,
oc cot %B cos 2 |6'+cot \G cos 2
oc (b + c)/(s—a), &c.
.UP
28. The Weill Point.
When an infinite number of m-gons can be inscribed in one
fixed circle, and described about another fixed circle, the mean
centre of the points of contact X, Y, Z, . . . with the inner circle
is a Jtaied point, which may be called the Weill Point of the
polygon (M'Clelland, p. 96). For a triangle the Weill Point is
G t , the centroid of XYZ.
In the triangle XYZ, since / is the circumcentre and Hi is
the orthocentre,
.•. G { is a fixed point on 01, and GjBj = 2 . GJ.
To determine the n.c. of G t .
Since 6?, is the mean centre of masses 1, 1, 1 at X, Y, Z,
.: a ol XYsin Z+ ZX bwY a: cos 2 %B+cos 2 %C.
29. We now proceed to discuss the loci of some well known
points related to ABC, which are poristically variable :
(a) The Feuerbach Point F,
(6) The Centroid G,
(c) The Orthocentre H,
{d) and (e) 0' the circumcentre, and I' the in-centre of
the Medial Triangle A'B'C.
20 MODEKN GEOMETRY.
Draw Hh, Gg, I'k parallel to O'l:
(a) The point V moves along 1 the in-circle,
(6) describes a circle, for OG — f . 00' ;
••• Og = %.OI;
thus g is fixed, and
Qg = §. O'l = i(/J— 2r) = constant,
(c) H describes a circle, for
OH = 2.00': .-. Oh =2.01,
so that /i is fixed, and
Hh = 2. O'l = E— 2r = constant,
(d) 0' obviously describes a circle, centre I, radius
(e) I' describes a circle.
For since I, I' are homologous points in the triangles A BO,
A'B'C, whose double point is G,
;. IGF is a straight line, and GI = 2.GT,
.-. Ik = § ,Ig, so that k is a fixed point, and
kl' = f . Gg = \ (E-2r) = 70'.
30. (/) The Nagel Point. — This point also belongs to the
series whose circular loci may be found by inspection.
Let XIx be the diameter of the in-circle which is perpendi-
cular to BO, and let the ex-circles I x , I.,, I s touch BC in X lt
GA in F 2 , AB in Z s respectively.
Then BX X = s-c, 0X 1 = s-b,
so that the equation to AX X is
y/(s-b) = ss/(s-o),
and thus AX r , BY 2 , CZ S concur at a point N whose b.c.'s are
as (s—a), (s—b), (s — c).
This point is called the Nagel Point of ABC.
If the absolute n.c. of N are aBy, then
aa/(s—a) = ... = 2A/s;
.•. a = \ . (s — a)/s.
Draw NP, NQ, NE perpendicular to AH lt Bfl„ GH S , then
AP = h x —a = \.a\s =s 2r.
.-. the perpendiculars from N on B X G V 0,^,, A^, the sides
of the anti-medial triangle A- i B 1 G 1 (...) are each = 2r.
Hence N is the in-centre of the triangle A l B^C l .
PORISTIC TRIANGLES.
21
The two triangles ABC, A-fi-fii have G for their common
centroid and centre of similitude, / and N for their in-centres,
and H for circumcentres, 0' and for Nine-Point centres.
The corresponding joins are parallel, and in the ratio 1 : 2.
Thus ON=2.0'I = B-2r,
and ON, O'l are parallel.
.-. N describes a circle, centre 0, radius = B—2r.
31. An additional note on this interesting point may here be
interpolated.
Since AB, AG are common tangents to the circles / and /„
and Ix, J,JT, are parallel radii of the circles, drawn in the same
direction,
.•. AxX} or Ax-NXj is a straight line.
And since AP = 2r = Xx,
;. PX is parallel to AxNX 1 ;
.-. PXX X N is a parallelogram, and PN = XX l = b — c.
Again XI = Ix and XA' = A'X 1 ■
.-. A'l is parallel to AN and PX.
Let IP meet BG in T.
22 MODERN GEOMETRY.
Then TH t : TX = TP : TI = TX : TA\
.-. TX i =TH l .TA'.
.•. T lies on the Radical Axis (common tangent) of the
in-circle and Nine-Point circle.
32. The Gergonne Point.
This is another point whose poristic locus is a circle.
Since BX = s — b and OX = s — c,
the barycentric equation to AX is y(s—b) =z(s—c), so that
AX, BY, GZ concur at a point whose b.c. are l/(s — a),
This point, the Centre of Perspective for the triangles ABO
and XYZ, is called the Gergonne Point, and will be denoted
by M.
To determine the absolute b.c. of M,
x _ 2A _ 2A" _ 2A 2
I/O — a) I/O - a) + ... ~ rj + rj + r, — 4R + r'
The join of the Gergonne Point Jf and the Nagel Point N
passes through H ( . (G.)
Proceeding as usual, the join proves to be
a(b— c)(s— a).x+... = 0,
which is satisfied by the n.c. of H^ which are (6 + c)/(s— a)
The join of M and O passes through o\ (G.)
For this join is {b — o)(s — a) .aa+ ... = 0,
which is satisfied by the n.c. of a, which are l/(s — a)
33. In the poristics of a triangle the following formulas are
often required.
(1) A = ra. (2) a&c = 4AE = 4Er.s.
Put s— a = s„ &c. . so that a = g t + »„ s = *, + »»+*,.
(3) *!«,«, = A 2 /* = r^s.
(4) l/^ + l/s.+ l/s, = (r ,+r,-l r s )/A = (4K + r)/rs.
(5) s 3 s s +... =s„s 3 s 8 (l/s 1 +...) =r (4B + r).
(6) A+-. = (",+»b) , «i+...
= (Sj + 8, + *,) (« 2 S 8 + S s Sl + gjg,) + 3SAS. • • •
= s.r(4E-|-r) + 3r 2 s = 4r (JS + r) .«.
PORISTIC TKIANGLES. 23
34. * The poristic locus of the Gergonne Point M is a circle
coaxal with 0(B) and I(r). (Greenhill)
Let 7T,, tt 2 , ttj be the perpendiculars from A, B, on the
Radical Axis JSyE^.
The power of A for the circle I(r) = (s — a) 1 .
But by coaxal theory this power is also equal to 2-rr^d ;
.'. it, = (s— a,y/2d, Ac.
But, if it be the perpendicular on the Radical Axis from any
point whose b.c. are (x, y, z), then
(x + y + z)ir = tt^x + ir 2 y + tt s z.
Hence, for M, whose b.c. are as 1/*,, &c,
(1/ Sl + l/s s + 1! s t ) ■ it = sffid . 1/s, + . . . ;
and, finally, *=-*-.*..
Again, the power II for the circle ABO of a point whose b.c.
are (x, y, z) is given by
n _ dyz + Wzx + Jay (QQ .
(x + y + zY
Hence, for M, whose b.c. are l/s lt ... ,
„ _ (a\+...)s l s. 2 s„ _ 4r(-B + r) ,
(V.+ .-)' (4E+r) 2 ' '
4tt+r
Or, the power of M for the circle ABO varies as the distance of
M from the Radical Axis. Hence M describes a circle coaxal
with 0(B) and I(r),
If m is the centre of this circle, then
II = 2. Om.Tr, by coaxal theory ;
., 0^ = ^+0 .A
4E+r
It may be shown that the radius of the M circle = -^-= ,
but the proof is long. •
* The original proof belongs to Elliptic Functions. For the proof here
given the present writer is responsible.
CHAPTER IV.
THE SIMSON LINE.
35. From any point Ton the circumcircle ABC, draw perpen-
diculars TX, TY, TZ to the sides BO, GA, AB. Produce TX
to meet the circle in t.
Then, since BZTX is cyclic,
I BZX = BTX or BTt = BAt ; '
.'. ZX is parallel to At.
So XY is parallel to At.
Thus XYZ is a straight line, parallel to At.
It is called the Simson Line of T, and T is called its Pole.
24
THE SIMSON LINE. 25
To draw a Simson Line in a given direction At.
Draw the chord tT perpendicular to BG, meeting BG in X.
A line through X parallel to At is tbe Simson Line required,
T being its pole.
Let AH V AO meet the circle ABG again in a, a'; it is required
to determine the Simson Lines of A, a', a.
(a) For A, the point X coincides with H v while Y and Z
coincide with A ; therefore AH X is the Simson Line of A .
(&) Since a'BA, a'GA are right angles, it follows that BG is
the Simson Line of a'.
(c) Drawing ay, az perpendicular to AO, AB. it is at once
seen that yz, the Simson Line of a, passes through H v and that
it is parallel to the tangent at A.
36. To prove that XYZ bisects TH {H orthocentre) .
If Q be the orthocentre of TBG,
TQ = 2E cos A - AH,
and QX = Xt = Ax,
since At, XYZ are parallel ;
.-. Hx=TX;
.•. HxTX is a parallelogram.
.-. XYZ bisects TH, say at h.
It follows that h lies on the Nine- Point Circle.
TOT' being a circumdiameter, prove that, when the Simson
Line of T passes through T', it also passes through O.
(W. F. Beard)
37. Let <t,, er 2 , <r s be the direction angles of XYZ, taking
BXZ = a v
To prove that the base angles of the triangles OAT, OBT,
OCT are equal to the acute angles which XYZ makes with the
sides of ABG ; i.e., to &„ <r 2 , <r s , or their supplements, as the
case may be.
Z OAT (or OTA) = frr—%.AOT
= frr-AtT = |ir- YXT = BXZ =
A relation of fundamental importance.
To determine QX or Xt,
Xt = Bt.Ct/2R,
and Bt — 2B sin BAt = 2R sin BZX = 2B sin <r s ;
.•. QX = Xi5 = 2iJ sin cr 2 sin <r 8 .
or..
26 MODERN GEOMETRY.
38. To determine the n.c. of T, the direction angles of XYZ
being o-,, <r„ cr s .
TA = 2E.cos OTA = 2E cos ^ ;
.-. a.2fi= TB.TG = 4E 2 cos o- 2 cos o- s ;
.■. a = 222 cos <r 2 cos <r 3 ;
so that the n.c. are as sec a- x : sec or 2 : sec o- 3 .
To determine the segments YZ, ZX, XY.
In the circle AYTZ, with AT as diameter,
YZ = AT sin .4 = 2E cos o-j . sin A — a cos o^.
39. To determine p, q, r, the lengths of the perpendiculars
from A, B, G on XYZ, the Simson Line of T.
From (37) Xt = 2B sin <r 2 sin <r s ,
and £AtX= ZXT = ^r-(r 1 ',
.'. £> = Xt . sin -4i.X = 25 cos o^ sin o- 2 sin <r s oc cot <r v
The equation to XYZ is therefore
COt CTj . x + cot o- 2 . y + cot <t s . z = 0.
To determine ir, the length of the perpendicular TO on the
Simson Line of T.
ir=TU=TXsm TXZ = 2B . cos o- 2 cos <r s cos <r,.
Or, since TA = 2S cos o-j,
p = TA.TB.TC/iB\
40. Let a parabola be drawn touching the sides of ABG and
having T for focus. The Simson Line XYZ is evidently the
vertex-tangent and U is the vertex.
Since XYZ bisects TH, the directrix is a line parallel to
XYZ and passing through H.
Kieperfs Parabola. — Let T be the pole, found as in (35),
of the Simson Line parallel to the Euler Line OGH. Let the
direction angles of OGH be 6 t , 2 , S , then
cos0j « (6 2 — c 2 )/a, &c. ;
so that the n.c. of T are a/(6 2 — c 2 )....
The directrix will be OffiT,
or (6 s — c 2 ) cos.A.a+....
THE SIMSOIT LINE. 27
41. Denote the vectorial angles OTA, OTB, 0T0, OTU by
<£n 4>v $s< 8 respectively, so that $„ c£ 2 , <£ 3 are equal to <r u <r 2 ,
o- s or their supplements.
To prove that 8 = ^, + ^j — <f> r
Since TtT = \v ; .-. arc Ci = BT' ;
.-. ^ «rc = srr = & ;
.-. T'Tt - T'TG—tTG = <£ s -4> 2 ;
.-. $«— 8 = TYZ = ZXT+ T'TX = (frr- *,) +*,-*, ;
.-. 8 = ^j + c^j— <^) s .
Making the usual convention that <f> s is to be negative when
falls on the side of TOT' opposite to A and 5, we may write
8 = ^ + 03 + ^,.
42. Consider the quadrilateral formed by the sides of ABO
and a straight line PQB.
The circles ABO, AQB, BBP, OPQ have the common point
M, and their centres 0, 0„ 2 , O s lie on a circle called the
Centre Circle, passing through M. (9)
Let 6 X , 6 2 , $ s be the direction angles of PQR.
If p x be the circumradius of AQB, the perdendiculars from 0,
on AQ, AB are p, cos B, p t cos Q.
So that, if a, B, y are the n.c. of 0„
/3 : y = cos i2 : cos Q = sec 2 : sec # 3 .
Hence A0 V B0 2 , G0 3 meet at a point N whose coordinates
are (sec V sec# 2 , sec0 3 ).
And, since a . -\ T -\ -r- — a cos 6 1 + . . . = ; (2)
sec l sec 2 sec 3
.•. N lies on the circle ABO,
and is the pole of the Simson Line parallel to PQR.
Again Z 0.N0, - tt-ANB
(BNOi being a straight line)
= v—C;
and since 00 2 , 00% are perpendicular to MA, MB,
.: 0,00, = AMB = ;
so that N lies on the Centre Circle, and is therefore the second
point in which the Centre Circle intersects the circle -ABC
28 MODERN GEOMETRY.
43. We will now deal with pairs of Simson Lines.
T and T' being any points on the circle ABC, it is required
to determine 8, the point of intersection of their Simson Lines.
Let o^o-jCTs, <ri'<r,'<7- 8 ', 6^8^ be the direction angles of the
Simson Lines of T, T', and of the chord TT'.
Then, if a, fi, y are n.c. of 8,
u. = SX sin (Tj = XX' sin o-j sino-j'/sin (o-j — o-/)
= TT 1 cos 0j . sin o-j sin o-//sin (o-, — o-/) .
But OJT = o- M OAT' = a-/, (M7)
so that TAT =■ a-,-0-,';
.-. T2" = 2£ sin (o-i-o-/),
and a = 2E cos 6 1 sin o-j sin <r/. (G.)
The equation to the chord which joins the points (a^y,) and
(a 2 /i 2 y 2 ) on the circle ABC is
aa/a^ + . . . = 0.
Hence the equation to TT' joining the points T, T' whose n.c.
are (secer,, ...)(sec<r/, ...) is
coso-j coscTj'.aa+ ... =0.
And the tangent at T is
cos 2 <r 1 .aa+ ... = 0.
44. To prove that the Simson Lines of ? and T', the ex-
tremities of a circumdiameter, intersect at right angles on the
Medial Circle.
THE SIMSON LINE.
29
Let these Simson Lines Xh, X'h' intersect at <d.
Produce TX, T'X' to meet the circle in t, f, so that TtT't' is a
rectangle, and tt' a diameter.
The Simson Lines of T, T' are parallel to At, At' (35), and are
therefore at right angles.
Again, since h, h' are mid-points of HT, HT', (36)
.-. hh' = \TT' = B.
But h, h' lie on the Medial Circle ; therefore they are the
ends of a Medial diameter.
Also hoih' = \tt, as shown above.
Therefore <o lies on the Medial Circle.
Since A'X = A'X', and XuX' = fjr,
.-. A'u - A'X or A'X' = B cos B v
where $ v 2 , 3 are the direction angles of TOT'.
30 MODERN GEOMETRY.
45. Let TOT' cut the Simsoti Lines mh, <ah' in k, V ; and cut
O'ti) in 0„ where 0' is the Nine-Point centre.
Since Hh = hT and Hh' = i'l",
.-. M' is parallel to TOT' or fcfc' :
And since hh' is parallel to kk', and O'/i = O'V,
.-. O-Jt = O-Je ; also ktok' = ^7r :
Hence kk' is a diameter of the circle described with Oj as
centre and O^ui as radius.
And since O'OjO) is a straight line, this circle touches the
Nine-Point circle at w.
46. Let o-/, <t 2 ', o- s ' be the direction angles of X'h'w, the Simson
Liiie of T>,
Then, since the Simson Lines of T and T' are at right angles,
so that the n.c. of T' are 2Bcos (-gTr— c 2 ) cos ( , §' ,r ~"" '3)> &'"••> or
2iJ sino-j sin cr 3 , &c. ; which are as coseco - ! : coseco- 2 : cosec cr 3 .
Or, since T't is parallel to BC,
;. T'X' = tX—2B sin o- 2 sin o- s .
Therefore the equation to the diameter TOT is
coso-j sin er 1 .a;+ ... =0.
47. Consider the figures ATOT', uXA'X'.
Since TAT' and X(oX' are right angles, while 0, A' are the
mid-points of TOT', XA'X', and
the angle OTA or OAT = A'Xm or A>wX ; (37)
it follows that
the figures ATOT', mXA'X' are similar,
a fact to be very carefully noted.
48. Through w draw Pu>p' perpendicular to BO.
Then from similar triangles AOp, <oA'p,
Op/R = A'p'/A'u = A'p'/B cos 6 1 ;
.■. A'p = Op cos X = perpendicular from p on 0.4' ;
.•. Po>p passes through p.
Hence, if Ap, Bq, Gr be perpendiculars to TOT', and pp', qq', rr'
be perpendiculars to BO, A, AB, then pp', qq', rr' are concurrent,
at that point co on the Nine-Point circle, where the Simson
Lines of T, T' intersect.
It follows that <« is the Orthopole of the diameter TOT' fsee
Chap. VI).
THE SIMSON LISE.
31
49. Since A'u> = B cos 0„ (44)
and A '<o = B sin A'Pw, in the Nine-Point Circle,
.•. A'Pto = ^tt — 6 1 — angle Opp' ;
.-. .4'P is parallel to TOT'.
Hence, to find w when TOT' is given, draw the chord A'P in
the Nine- Point circle, and then draw the chord P<o perpendicular
to BO.
50. The A'B'C n.c. of w.
From (35) the Simson Line of <d in the Nine-Point circle
is parallel to A'P, and therefore to TOT'.
Hence the direction angles of this Simson Line are 6 V 8,, 8 S .
It follows from (38) that the A'B'C n.c. of o> are
a' = 2 . \B cos 6 2 cos 6., <fcc,
which are as sec lt sec 2 , sec 6 S .
The ^IBC n.c. of w.
From the similar triangles iaA'p', A Op,
a = o>p' = *4'<o . _4p/J = p cos 0J.
Hence the absolute n.c. of <d are p cos #„ q cos <? 2 , r cos 8 a .
To determine a in terms of p, q, r.
32 MODERN GEOMETRY.
Let p', q', r' be the perpendiculars on TOT' from A ', B', 0".
Then since A' is mid-point of BO,
.'. 2p' = q + r;
also p' = A'O. cos 6 1 = B cos J. cos 6 l ;
••■ « = li q + T \ S ■•• aa = 1 »(g- + r)tanX.
2/c cos -A
The formulas of (14) and (50) supply us with a very simple
proof of (8).
For aa = bp'+cy l . (14)
.■. a.p cos 8 1 = b.R cos 8 S cos 6 1 + c.R cos 0j cos 6, ;
.'. ap/R = 6 cos & s +c cos 2 .
51. To illustrate the use of these formulas, take the case
when TOT' passes through I, the in-centre.
Here cos^ = \{b— c)/d, p = R/d. (6-c)(«— a) /a [d=OI].
So that 4a' = R(c-a)(a-b)/d* cc l/(b—c),
and aa = ^R/d*. (b — c) 2 (s — a) cc (6 — o) 2 (s— a).
Hence <o is the Feuerbach Point.
52. To determine the A'B'O' n.c. of 0^ the centre of the circle
in (45).
Taking A'B'O' as triangle of reference, the equation to TOT'
is cos Acos6 1 .aa'+ ... = 0. (16)
The equation to the diameter of the Nine-Point circle passing
through o>, whose n.c. are (sec^, ...) is
sin 0, cos 6 X . aa' + . . . = 0. (46)
Hence at V where these lines intersect,
aa' . cos 6 X cc cos B sin 3 — cos sin 2
a cos 7? (p — q)/sin 0— cos (r—p)/ sin B ;
.-. a'cos#i ocp (sin 21? +80120)— ?. sin 2B— r sin 2 (7.
But since TOjP' is a diameter of ^LZ?0,
p sin 2^4 + g sin 2B+r sin 20 = 0.
Hence a' cc p sec 0„ &c.
To determine the radius O^ui (= p) of this circle.
The perpendicular tt from w on the A'B'C Simson Line of u>
= 2 . %R . cos 6 l cos 6, cos S . (39)
The angle h' which this makes with the diameter aiOft'
= «, + «, + (»,. (41).
THE SIMSON LINE. 33
The perpendicular from u> on TOT'
= 2 perpendicular on Simson Line = 2ir ;
.•. p = Ojco = 2tt sec 8' = 2. B cos tfjcos ft, cos S . sec (Oi + O^+O^).
The parabola, which touches the sides of A'B'C and has <d for
focus, has the Simson Line of <o for its vertex tangent, and for
its directrix the diameter TOT', which is parallel to the Simson
Line and passes through the orthocentre of A'B'C.
53. The envelope of the Simson Line is a Tricusp Hypocy-
cloid.
Since /oJ'JT 1 = AOT,
.-. mO'H 1 -2.AOT.
But Medial Radius = %R ;
.•. arc -HjO) = arc AT = 2 arc dh,
by similar figures Hdh, HAT.
Now take arc A'L = -| arc A'S V
and draw Medial diameter LO'l.
Then, since A'd is also a Medial diameter,
arc A'L = arc ZtZ ;
.•. arc S t L = 2 arc A'L = 2 arc Ze?.
Also arc fljiu = 2 arc Tirf ;
.■. arc Lo> = 2 arc Ih.
Therefore Xiah touches a Tricusp Hypocycloid, having 0'
for centre, the Medial Circle for inscribed circle, and LO'l for
one axis.
54. Let DOB 1 be the circumdiameter through I, the in-centre.
Let fa, fa, fa denote the vectorial angles B'BA, B'BB, B'BG.
Draw the chord OIF.
Then P is the mid-point of the arc AB ; and
.-. lFBB' = %(fa + fa).
Also IPD' or CPD' = ODD' = <£ 3 ;
.-. IPD = ^ir-fa.
Let r/jR = to ; then
l±m_W_W IP __ sin iPD' sin POP'
l_ m ~ ID ~ It' IB sin BB'P ' sin 7PP
sin £,, sm^(0i + ^» 3 ) .
cos | (0! + fa) ' cos fa
D
34
MODERN GEOMETRY.
tan \ (fa + fa) = = — — . cot fa.
Put fa + fa—fa = &; sinSsS, sin fa = s.
Then S = sin (fa + fa) cos fa — cos (fa 4- fa) sin fa
- 2 tan £(.*>', + <£,) , l-tan'IQk + 6) . .
.-. S/s. {(l-m)"' sin* <b a +\l+my cot? &}.
— 2(1 — m 2 ) cos 2 <jfc 3 — (1— m) 2 sin 2 <£ 8 4-(l + m) 2 co8 2 <£ s ,
And finally,
S/s. (l + m) 2 -4ms 2 } = (3— m)(l+m) — 4s 2 ,
or s s -mS'..9 2 -i(l + m)(3-m).s + i(l+m) 2 .S = 0.
B D
From (42) the angle 8 is the vectorial angle of the perpendi-
cular from B on the Simson Line of B, and this Simson Line
passes through the Peuerbach Point F, and through /, one of
the fixed points, where the circle I(r) cuts the axis BIOB' (45).
The cubic then gives the values of fa, fa, fa, the vectorial
THE SIMSON LINE.
35
angles of DA, BB, DO ; and thus the triangle ABO is deter-
mined for any given position of F.
Again, if we put cos 8=0, cos <£ 8 = c, we shall obtain
c , -m0.c ! -i(l-m)(3+m).c-l(l-rf).0 = 0.
(Greenhill).
55. In a poristic system of triangles ABO, the Simson line of
a fixed point 8 on 0(B), passes through a fixed point.
Let B8 V S/Sj be tangents to I(r) ; then iS 1 S 2 is also a tangent.
Draw DE touching I(r) and parallel to 8,8 ^.
Draw 8T perpendicular to DE.
Then shall the Simson Line of 8 pass through the fixed point T.
Let BE be cut by AB, AG in y, B.
Let AC, AB cut S, S, in F, G ; let AS„ AS, cut DE in H, K.
Now IB, IE, Iy, IB are perpendicular respectively to IF, IS 2 ,
IG, I8 T ; and hence
(BEyB) = (F8.GS,) = (BHyE).
BE_ yB_BH yK
BB' yE BK'yH'
BE. BE _ y E.yK
Thus
BD.BH yB.yE
Again, L SBH = SS& = SA8 2 or SJ.H.
.•. SDAH is cyclic, and similarly SEAK,
36 MODERN GEOMETRY.
The relation (i) shows that the powers of ft and y for these
circles are in the same ratio ; and hence that ft, y lie on a circle
coaxal with these circles, and therefore passing through A and 8.
That is, ASfty is cyclic.
Now the four circles circumscribing the four triangles formed
by AB, AG, BC, fty have a common point, which must be S,
since this point lies on the circles ABO, Afty.
Hence d-jo^cr, and T, the feet of the perpendiculars from S on
the lines BC, CA, AB, fty, are collinear.
In other words, the Simson Line oyr s cr, of S for the triangle
ABG passes through T, a fixed point, being the foot of the
perpendicular from S on the fixed line /3y.
(Grreenhill and Dixon)
The properties given in (34), (54), (55) present themselves
in the Cubic Transformation of the Elliptic Functions. v
CHAPTER V.
PEDAL TRIANGLES.
56. From a point 8 within the triangle ABO draw Sd, Se, Sf
perpendiculars to BG, GA, AB respectively, so that def is the
Pedal Triangle of 8 with respect to ABC.
Let angle d = A, e = /*, / = v.
Produce AS, B8, G8 to meet the circle ABO
L, M, N.
From the cyclic quadrilateral SdCe,
L Sde = SOe or NGA = NLA,
so Sdf = ML A ;
.-. dov\ = MLN, &o.
Thus LMN is similar to the pedal triangle of 8 with regard
to ABO.
Similarly ABC is similar to the pedal triangle of S with
regard to LMN.
.: B80=BM0+S0M
= BMO+NOM or NLM = A + \.
To determine a point 8 whose pedal triangle has given
angles A., p, v, describe inner arcs on BG, GA, AB respectively,
containing angles -4 + A, B+/*, 0+v, any two of these arcs
intersect at the point S required.
37
38 MODEBN GEOMETRY.
This construction also gives S as the pole of inversion when
ABC is inverted into a triangle LMN with given angles A., /*, v.
57. If r„ r s , r, represent SA, SB, SG, the tripolar coordinates
of 8, and if p be the circumradius of def,
then 2p sin A = ef = SM. . sin e^f
(in circle SeAf, diameter AS)
= rj sin A ;
.-. M JV : WL : LM = ef :fd : de = sin A. : sin /u. : sin v
= ar^ : br t '. cr v
Limiting Points. (G.)
In section (21) let TA, TB, TC meet the circle ABC again
in LMN.
Then from (57) MN oc sin A or a»v
In (a), TA a: s— a; .'. MN cc a(s—a).
In (6), TAxl/Ja; .: MN <x a/ Ja <x */a.
In (c), T-4aca; .-. MN o: a?.
In (d), T.A a -/cot J.; .-. JfJVocsin^ s/cot^ oc v/sin2J..
Let J x , X, be the in-centres of LMN in (a) and (c) ; ff 2 , ff,
the orthocentres in (6) and (d).
It is very easy to prove that
Olt = 01, 01, = Off, Off, = 01, Off 4 = Off.
58. In (21), since
p : q : r <x sin A/a : sin /j./b : sin v/c,
the Radical Axis becomes
sin 2 A/a 2 . x + sin 2 /n/6 2 . y + sin 2 v/c 2 . z = 0.
To determine J, the pole of this Radical Axis for the circle
ABC.
The polar of a'/8y is (6y'+c/8')a+ ... = 0.
.-. sin 2 A/a oc &■/ +■ eft ;
.'. sin 2 A a acfi' + aby' ;
;. bca! oc — sin 2 A+sin i y + sin 2 v oc — ef+fcP + de*
a cos A .fd . de oc cos A . sin /i sin v
oc cot A;
.•. a'oc acotX;
so that the n.c. of / are a cot A, 6 cot /u, c cot v. (Dr. J. Schick)
Z7/A = ,.
PEDAL TRIANGLES. 39
59. To show that J7, the area of def, is proportional to II. the
power of S for the circle ABC.
2 U = de . df. sin \ = r 2 r 8 . sin B sin sin X,
and 2 A = 4E 2 sin 4 sin B sin 0.
t r 2 r a sin A,
~ IP 'sin A'
Now r 2 .SM=LT;
and in the triangle SMC,
r s or SG = 8 M. sin A/sin \.
Hence U/A = An/B 2 = £ (E 2 - OS 2 )/i?- 2 .
When U" is constant, OS is constant, and S describes a circle,
centre 0.
Let a, B, y be the n.c. of S.
Then £y sin A + ... = 2(AeS/+ ...)
= 2Z7=i.(E 2 -08 2 ).A/B 2 .
Putting 4A 3 = (oa + 6/3 + cy ) 2 ,
we have a/3y+... = (IP-OS^/abc. (aa + bB + cy) 2
as the locus (a concentric circle) of S, when OS is constant.
When OS = B, U vanishes, as the pedal triangle becomes a
Simson Line, and aBy + bya+caB = 0.
60. To determine the power II of a point 8 in terms of x, y, z,
the b.c. of 8.
Since aa/x = bB/y — cy/z = 2A/(x + y + z),
.-. a = 2A/(x + y + z).x/a, &c. ;
„„ n . . , 4A 2 ( y z a , )
(:t!+2/ + z) z (6 c 2E )
.-. n or (fl 2 - OS 2 ) = 4B 2 /A . 17 = «V +*»'»"+ Ay .
(a+y + z) 2
Note that only the ratios x : y : z are required.
Examples. —
(i) For I, z <x a;
n = _^_ = 2Br.
a + b + c
40 MODERN GEOMETRY.
(ii) For B, x <x tun A ;
a- tan B tan G+ ...
~~ (tan A + tan B + tan O)' 2
= 8 ft 2 , cos A cos B cos 0.
(iii) For C?, a; = y = z ;
.-. n = i(a a + 6 2 + c 2 ).
61. When S lies on a known circle for which the powers of
A, B, G are simple expressiops, the power II of S for the circle
ABO usually takes a simpler form than that given by the above
general formula.
Let d be the distance between the centres of the two circles
Q and ABG ; t^%% 2 the known powers of A, B, G for the circle Q ;
mi-jTrjTr,,, the perpendiculars from S, A, B, G on the Radical Axis
of the two circles.
From (7), we have rr = *i»+»iy + »i' ,
v ' x + y+z
But, from coaxal theory,
i, 2 = Zd.ir,, &c, while II = 2d.ir ;
._ n _ t*x+t*y + t*z ^
x + y + z
Equating this to the expression for II, found in (60 )
we have aryz 4- Vzx + c*xy = (t*x + tfy + i 3 2 z) (x + y + z)
as the locus of 8 : i.e. the circle Q.
As an example, let us find EC for the point <u (50).
This point lies on the Nine-Point Circle, for which the power
of A = t* = \ be cos A, &c.
Also for hi, x = aa = p (q + r) tan A, (50)
and x + y + z = 2A.
.-. n= {%bccosA.p(q + r)ta,nA + ...\/2A,
= qr + rp+pq.
62. The Radical Axis of the circles Q and ABG is
t*x+t*y + t>z = 0,
for 7r, a *, 2 .
Examples. —
(i) When Q is In-circle : t* = (s-af.
(ii) When Q is Circle 7,1,7, : i, 2 = be.
PEDAL TRIANGLES. 41
(iv) When Q is Anti-medial Circle (1) : t? = a 2 .
(v) When Q is Nine-Point Circle : *,' = ■§ be cos A oc cot A.
(vi) When Q is circle 2\T.,2' 3 , circumscribed to the triangle
formed by the tangents at A, B, G :
*, 2 = E 2 tan £ tan C oc cot A
(vii) When Q is the circle (£H) : i, 2 = 2E cos A.%h,cc cot A.
(viii) When Q is Polar Circle :
t? — AE^+^R* cos A cos B cos a cot A.
So that circles (v) (vi) (vii) (viii) have the same Radical Axis
cot yl. a; + cot B.j/-)- cot O.z = 0.
63. Feuerbach's Theorem.
To determine the Radical Axis of the Nine-Point circle and
In-circle :
Take A'B'C as triangle of reference,
The power of A' for the In-circle = L* = A'X 1 — %{b— c) 2 ;
.'. the Radical Axis is (6— c) 2 aj'-)- ... = 0.
But this is the tangent to the circle A'B'C at the point whose
n.c. are 1/(6 — c)....
Hence the two circles touch, and 1/(6 — c)... are the n.c. of
the point of contact.
64. To express II in terms of A., //., v.
We have L BSG = A + A ; (Fig., p. 37)
.-. aa = 2. ABSORBS. SO. sin(A+\)
=*= BS.SM. sin (A + A.) sin A/sin X
= n {sin 2 4 cot A +£ sin 24} ;
.'. 2A/nor (aa + 6/84-cy)/n = 2. sin 2 J. cot \+%%.sm2A.
Multiply each side by 4B 2 . Then
8 E 1 A/11 or 2B . abc/Tl — a 2 cot A. + 6 2 cot /x + c 2 cot v + 4 A
= if.*
.-. TIM = 2B . abc or SJS 2 A, II = 2B . abc/M ;
giving II in terms of A, fi, v.
* The expression "ffl 2 cotA+..." was first used, I believe, by Dr. J.
Schick, Professor in the University of Munich. The relations he deals
■with are different from those treated in this work.
42 MODERN GEOMETRY.
Since B' - OS 2 = II = 8 R 2 A/ikf ;
. offl 8= ^coU+...-4A
' a 2 cotX+...+4A
65. Observe also that the area of def
= TT = A. A/E 2 .II = 2A 2 /M";
so that, if p be the radius of the pedal circle def,
2f. sin X sin /*. sin v = U = 2b. 2 / M ;
and now all the elements of def are found in terms of X, /x, r.
From above, aa = II . sin (4 + X) sin -4/sin X.
afec sinf^+X)
Hence
Af sinX
66. To illustrate one of these results, take S, the focus of
Artzt's Parabola, which touches AB at B and AG at (see
Figure, p. 88). (G.)
It is known that I SAG = SB A, SAB = SO^L ;
.-. BSK, = BAS + ABS = BAS + CAS = A.
So G8K^-A.
.: BSA = it- A = B+G= GSA ,
and BSC = 2A.
So that, if X, [i, v are the pedal angles of S,
A + \ = 2A; B+ f i = B+G; C+v = B + C;
.-. \ = A,/jl=G,v = B.
.-. M = a 2 cot^ + 6 2 cot(7+c 2 cotB + 4A.
Putting 4A.cot4 = — a a +6 2 +c 2 , &c,
we obtain > 2A.M = a 2 (2& 2 +2c 2 -a 2 )
= 4a 2 m a 2 , where m l = AA'.
Then jp 2 sin A. sin sin 5 — - A*/M ;
and finally p = i&c/m,.
Then S4 .sin J. = e/ = 2p sin X ;
.-. SA = l-fec/m, ; so SB = |.c ! / Wl , SO = \.h*lm v
The n.c. of 8 can be at once obtained from
a = abc/M. sin ( 4 + X) /sin X, &c.
67. If, with respect to the circle ABG, the point 8 X be the
inverse point of S, lying on OS produced; then OS.OSi = I? 2 ,
PEDAL TRIANGLES.
43
so that S, St may be taken as the limiting points of a coaxal
system, to which the circle ABG belongs.
If 08 S x cuts this circle in T, T\ then AT, AT' bisect the
angles between AS and AS„ &c, so that
S r A : 8A = S,T : 8T = S,B : SB = S,G : SG.
Let (Lfi-ifi be the pedal triangle of 8 X ; then in the cyclic quad-
rilateral SfrAfi . ej/i = S X A . sin A ;
■'■ e i/i : fA '■ d r e x = /S'^.sin^. : SJi.sinB : SjO.sin G
= SA .sin A : SB .sin B : SG .sin G
= ef : fd : de.
Hence the pedal triangles of the inverse points S and 8 X are
inversely similar : so that
d, = \, gj = //., /i = v.
To prove that the triangle LiM^N^ is similar to c^e,/,, the
pedal triangle of /Sj.
In the cyclic quadrilateral S^ce^,
/ /S,d,ei = 8 1 ce 1 = n—N^GA = N^A.
So S.d,/, = ^,-M-
.•. e^/j or A. = M^LjNj, &c.
Similarly ABG is similar to the pedal triangle of S x with
respect to the triangle L 1 Jf l .W].
To determine #, first find 8 (56), and then obtain S x as the
inverse point of S.
44 MODERN GEOMETRY.
We now have a second pole ($,) from which ABG can be
inverted into a triangle L I M 1 N{ with given angles X, /a, i>.
68. To express IIj, the power of ^ for the circle ABG, in
terms of X, //., v.
lBS,C = BM.O-M.CS, (MjOJVO
= BM l C-M 1 L 1 N 1 [cyclic quad. L^CN^
= A-X.
So AS,B = G-v.
Also ^£,0 = AS.B + BS.G = (A-X) + (0-v)
= fi-B. [> > B]
.: aa, = 2 . AB^C = B^ . CS, . sin BS,G
= BSi . S^ . sin (A — X) sin ,4/sm X
= II, . sin (A — X) sin .4/sin X. ,
So cy, = rTj . sin ( C—v) sin 0/sin v,
and 6/8, = TTj.sin (B—fi) sin B/sin/j.,
giving y8, a negative value [/x > B], as the figure indicates.
Proceeding as before, we find
n^ = ZR.abc or 8fi a A, where M 1 = a 1 cot X+ ... -4A,
giving FEj in terms of Jfn and therefore of X, «,, v.
Since n, = OS? -IP = 8B 2 A/Jf„
. OS 1 2 _ a 8 cotX+...+4A
ii 2 a 2 cotX+...-4A"
[or from OS. OS, = E 2 ]
Hence OS'VE 2 = Jf,/M, and OS/AH' 2 = M/M 1 ;
ST _ R-0S VM- VM,
ST' B+0S~ VM+VM X '
The area of A^e,/, = #i = 2A7.M,.
Hence 1/U-1/I7, = 4/A.
If Px be the radius of the pedal circle d 1 e l f 1 ,
2p, 2 . sin X sin p sin v = Z7 t = 2A i /M l .
And now all the elements of c^e,/, are found in terms of
From above, aa^ = II, . sin {A — X) sin .4/sin X.
Hence a, = «*£ . sin ^~^),
AT, sin X
PEDAL TRIANGLES.
45
69. In section (47) it was proved that the figures A TSOT and
taXdA'X' are similar, and just as S is homologous to d, so is S,
homologous to d,.
Now AT bisects the angle 8AS 1 ;
.-. o)X bisects the angle dwd v
It follows that i
ud : <od, = Xd : Xd 1 = TS : TS V &c,
so that od : we : <of = cod!! : u>e, : o>/i.
Hence <o is the double point of the similar figures ; and <oX,
ioX', the Simson Lines of T, T', are the axes of similitude for the
inversely similar triangles def and d,e 1 / 1 . (Neuberg)
CHAPTER VI.
THE ORTHOPOLE.*
70. Let p, q, r be the lengths of the perpendiculars Ap, Bq, Or
from A, B, G on any straight line TT', whose direction angles
are B v 6 2 , 6 S .
Draw pSj perpendicular to BC, qS 2 perpendicular to OA,
rS s perpendicular to AB.
These lines shall be concurrent.
For B8 l = BD+p sin 9 V OS, = OD—p Sin 0„
and 2.apsintf 1 = 0; (2)
.-. %( k B8 1 , -GS l s ) = 2(J5D 2 -(7D 2 ) + 2.2a^sin^ = 0.
Hence S^, S^q, S s r are concurrent.
The point of concurrence, denoted by S, is called (by Professor
J. Neuberg) the Orthopole of TT-
The Orthopole theorems are nearly all due to Professor J. Neuberg.
46
THE OETflOPOLE. 47
Let 3 be the length of the perpendicular SS' on TT'.
Now, for this one particular case, take V 3 , 6 S to represent
the acute angles which the sides of ABG make with TT'.
We have pq = c cos 6 S .
Also pSS' = 0, ;
since Sp, SS' are perpendicular to BG, TT' ;
also qSS' = 2 and pSq = G ;
since Sp, Sq are perpendicular to BG, GA.
.-. 8p = c cos 3 .sin (90°— 2 )/sin G = 2R cos 2 cos 3
and 3 = Sp cos t = 2B . cos 6, cos 2 cos 6),.
As T2" moves parallel to itself, the figure SqS'pr remains
unchanged in shape and size.
71. To determine the Orthopole geometrically.
Let Ap meet the circle ABO again in B.
Draw the chord BB'B" perpendicular to BG.
Let BA, GA meet TT' in r\ r".
Then BE = 2E. sin {BAB or r'Ap) = 2B cos S ,
so GE = 2E cos 2 .
.-. BB = BB. GB/2B = 2B cos 2 cos 3 = Sp.
Hence S is found by drawing B'S, pS parallels to Ap, BB'.
As TT' moves parallel to itself, 8 slides along B'S perpendi-
cular to TT'.
Also B'8'S, being parallel to AB, is the Simson Line of B".
72. To determine the ABG n.c. of 8, (G.)
u. = 88 x = projection of BqS on SS}
= q cos <?! + Sq cos pSq
= q cos a + 2E cos 6 1 cos S . cos G.
.'. asec0! — 2 + 2fi cos (cr — ap cos B—bq cos A) /2\
from (2).
Multiply first term on right side by sin A sin B sin and the
other terms by 2A/4E 2 .
Then
a sec 0j . A/2E 2 = sin 4 sin C . q sin 5— cos J. cos G . q sin B
+ cos (7 . r sin G— cos B cos ,p sin ^
= cosB.g sinE+cosC.7-sinG'— cosEcos G.psinA.
48
MODERN GEOMETRY.
Also 2A .cos 0, = op— cos . bq— cos B . cr. (2)
*.-. a = J2/2A 2 . (ap—bq cos G—cr cos B)
x (cos 77 cos G . ap— cos B.bq — cos Cor).
Multiplying the two factors, and using the form
2.ay-S.gr.2focos,4 = 4A 3 , (5)
we obtain
2Aa = abc cos 73 cos G+ aqr —rpb cos G—pq . c cos B.
When TT', whose equation is ap.a+... = 0, passes through
the circumcentre 0, then ap . cos A+ ... = 0.
In which case
a = E/2A 2 . 2A cos 0j . (cos B cos C.ap + ap. cos ^1)
= £> cos 0,.
73. Let <Oi' be the circumdiameter parallel to TT'.
Then cr, the orthopole of tt', lies on the Nine-Point circle.
Let 7J be the orthocentre of ABG.
Since Air = ■n-J?, .'. Ji'o-! = o-jJJ,.
Again Ji"E' = RR' + 2 . CM' = a-n- + A H ;
.-. R"R' + HH 1 = a7r+AE v
* It was Mr. T. Bhimasena Rao who first pointed out that the
expression for each coordinate involves two linear factors in p, q, r. His
method is different from that given above.
THE ORTHOPOLE. 49
But Air' = BR' (equal triangles Amr', BwB') = o-ir.
Aud ff'jBTi = 2.tt<t,;
.-. B"B' + HH, = 2(cnr + w, ) = 2 . era-,.
Heace o- is the mid-point of HR", and therefore lies on the
Nine-Point circle.
74. To find a /3 y , the ABG n.c. of <r. (G.)
Bisect ir'jff - ! in m.
Then o-7r = 4tt' and -k(t x = tt'tm.
a = (Tcr l = Am = Air cos mJ.Tr = p cos 6 X , &c,
as found in preceding article.
To find a '/8 V„', the _4'5'<7 n.c. of <r. (G.)
Since tra-j is = and parallel to Am,
.". A<r is = and parallel to o-jTO.
Also iro-j^m is a rectangle ;
.•. AH-jiTo- is a symmetrical trapezium.
Hence the circle (Jm), passing through H l and jt, passes also
through o\
.-. 90° — o-FjA' = crH^A — <rirA = e i ;
.-. a A' = E sin o\Hi-4' = Zi cos 6, ;
.'. a ' = <rB' .irC'/B = It cos 8. 2 COS 3 , <fcc,
so that ar is (sec fl 1; sec 2 , sec S ) referred to A'R'C-
It follows that the point <o of sections (44-50) coincides
with the Orthopole o- of a circumdiameter TOT'.
75. The Simson Lines of the extremities of any chord TV
of the circle ABG pass through 8, the orthopole of TV-
For, since L TpB = 90° = Ti^E,
.". TTt-ipR is cyclic ;
.-. pR 1 X or ^S/A = TBp or !ML4 = 2T,A ;
.-. BiP is parallel to AT V
But ^-Bi = R'B = Sp ;
.-. XS is parallel to Eip, and therefore to T^A.
Hence XS is the Simson Line of T.
So for V.
50
MODERN GEOMETRY.
Thus we have three Simson Lines SX, SX', SB', all passing
through S, their poles being T, T', B" respectively.
They are the three tangents drawn from 8 to the tricusp-
hypocycloidal envelope of the Simson Lines, so that each of the
points T, T', B" has similar relations to the other two ; for
example :
(a) Just as B'S is perpendicular to TT', so XS is perpen-
dicular to T'B", and X'S to TB" ; or each Simson Line is per-
pendicular to the join of the other two poles.
(&) As p, the foot of the perpendicular from A on TT', lies
on the perpendicular to BG from 8 : so also do j3, and p 2 , the
feet of perpendiculars from A on TB", T'B".
Hence ppip 2 is the Simson Line of A in the triangle TT'B".
Thus (S. Narayanan) the Simson Lines of A, B, G for th
triangle TT'B" as well as the Simson Lines of T, T, B" for AhG
all pass through 8.
THE ORTHOPOLE.
51
76. Let TT' cut BO in U, and let the circle (AU), passing
through H x and p, cut Sp in p'.
Then Sp . Sp' = twice product of perpendiculars from and
S on TT'. (G.)
In the circle (AU) the chords pp', H^A are parallel;
.'. the trapezium Ap'pH^ is symmetrical.
But the trapezium Aa-n-H^ is also symmetrical ;
.'. p'cr = pir = <2 and jpr = So" ;
.'. Sp' = 2Scr cos a-Sp' — 2d cos 0J.
But Sp = 2B cos 2 cos 0, = 3 sec 6 X ;
.-. Sp.Sp' = 2d8.
Note that 2d& is therefore the power of S for the circle (AU).
Similarly, if TT' cut GA in 7, AB in 17, the power of
S = 2dS for each of the circles (BV), (017).
Therefore in the quadrilateral formed by BG, GA, AB, TT',
the orthopole (S) of TT' lies on the 4-orthocentre line, or
common Radical Axis of the three diameter circles.
77. Lemoyne's Theorem.
If P be any point on the straight line TT', whose orthopole
is S, then the power of N with regard to XYA, the pedal circle
of P, is constant.
52 MODERN GEOMETRY.
Draw pB, py parallel to BB, EG, so that the figures ABpy,
ABBC are Nomothetic.
Draw AX' parallel to BG, XPX' perpendicular to BO.
Let YZ cut By in x.
Since E lies on the circle ABC, therefore_p lies on the circle ABy.
Lpyx or pyfi = pAB or pAZ
= pYZ or pYx
(circle AYpZPX', diameter AP) ;
Yxpy is cyclic ;
.-. pxy = pYy = 180° -pYA = pXJ. ;
.'. pa;.X' is a' straight line.
Let Xx cut pS in 8.
Then Sp/XX' = px/xX'
— ratio of perpendiculars from p and A on /3y
= ,, ,, „ B and .4 on BG
(from similar figures ABpy, ABRG)
- BB'/XX' ;
.-. &p = BB'=:pS;
.'. 8 coincides with 8.
Observe that, since E^pp'A is symmetrical,
.■. Xpp'X' is symmetrical, and therefore cyclic.
Denote the circles XYZ, XX'pp, AYpZPX' by L, M, N
respectively.
Let L and M intersect again in a'.
Now the common chord of L and M is Xx'.
„ „ „ M and N is X'p.
„ „ „ L and JV' is F.Z.
These common chords are concurrent.
But X'p, YZ pass through x ;
.'. Xx' passes through x, and therefore through S.
Finally, since Xpx'p'X' is cyclic,
.-. sx: &' = s P . Sp' = 2ds.
But X, x' are two points on the circle XYZ.
Hence the power of 8 for the circle XYZ = 2dS.
Note that the circles {ATJ), (BV), (CW) are thepedal circles of
u,v,w.
78. When TT' is a circumdiameter tOt', then d vanishes, and
then the pedal triangles all pass through the orthopole a- of tOt'.
THE ORTHOPOLE. 53
Since in this case At — ttB, the dimensions of the nomothetic
figures ABiry, ABBG are as 1 : 2, so that /8y becomes B'C.
Hence Xcr and YZ intersect on B'C.
79. Returning to the general case, the diagram shows that if
x, y, z are the b.c.'s of S with regard to the triangle XYZ, then
x _ ASYZ _8x = SP
x + y + z AXYZ xX XX
_ 2B cos 2 cos 0, .
2B sin B sin '
.■. x : y : z = sec #, sin A : secft, sin B : sec# 3 sin C.
Thus the Orthopole has constant b.c. for every one of the
pedal triangles XYZ. (Appendix I.)
80. In section (20) we determined the inverse points T, T'
whose tripolar coordinates are p, q, r by dividing BG at P, P' ;
OA at Q, Q! ; AB at B, B' ; so that BP : PC = q : r, Sec., and
describing circles on PP', QQ', Bti 1 .
Conversely we may begin by taking two inverse points T and
T', whose tripolar coordinates are p, q, r. Then the internal
and external bisectors of BTG, (BT'G) meet the sides in the
points P, P', &c. ; for
BP : PG= BT : TC = q : r, &c.
It is known that the triads of points P'Q'B', P'QB, Q'BP,
B'PQ lie on four straight lines, the equation of P'Q'B' being
px + qy + rz = 0,
while that of P'QB is — px+qy+rz = 0, &c.
To prove that the point common to the four circumcircles of
the four triangles formed by these four straight lines is w, the
Orthopole of OT'J"
Describe a parabola touching P'Q'B' and the sides of the
Medial Triangle A'B'C', and take A' B'C' as triangle of reference.
From (15) the A'B'C equation of P'Q'B' is
(q + r)^ + {r+p)y'+(p + q)z' = 0.
Then from (9) the n.c. of the focus are as - TT -. K , &c,
p (s —n
where \_q + r = p'], ■■-, or as -j
a
q'—r-
Changing +p into— p makes no change in the focus, therefore
the parabola also touches P'QB, and similarly Q'BP and B'PQ
54
MODERN GEOMETRY.
Wow let 6„ 6.,, # 3 be the direction angles of OTT', and draw
Trn perpendicular to BC.
Then g 2 -r 2 cc BT'—CT 2 oc 2a. Am oc 2a.O2 T cos0 1 ;
a a
■■ -. § « sec 0,.
Hence the orthopole <o, whose n.c. are (sec0„ ...) is the
focus of the parabola : the Simson Line of u> (in A'B'C") being
the vertex tangent, and OTT' the Directrix. And, the circum-
circles of the triangles formed by any three of the four tangents
P'Q'JS', ..., pass through the focus u>.
On the fixed Directrix TOT' may now be taken an infinite
number of inverse pairs (TT 1 ). For each pair we have a set of
four harmonic lines touching the parabola, the circumcircles of
the four triangles passing through w, and the four orthocentres
lying on TOT.
In addition to each set of four harmonic tangents, there are
also the three tangents B'O', G'A', A'B'. The student may
develop this hint. (G.)
Remember also that the pedal circles of all points T or T' also
pass through to. {Appendix II).
CHAPTER VII.
ANTIPEDAL TBIANGLES.
81. If S be any point within the triangle ABG, the angles
BSG, C8A, ASB are called the Angular Coordinates of 8 ;
they are denoted by X, Y, Z.
If A., p, v are the angles of the pedal triangle of S, then
X= BSC = A+\.
So that, if a, /8, y are the n.c. of S,
cm = II . sin (A+X) sin .4/sin X (65)
= II . sin X sin .,4/sin (X— A) .
82. Orthologic Triangles.
N
Draw any straight lines MN, NL, LM perpendicular to 8 A,
SB, 8G, so that the perpendiculars from A, B, G on the sides
of LMN are concurrent (at S). Then shall the perpendiculars
from LMN on the sides of ABG be concurrent.
Let AS, MN intersect at Z, &c.
55
56 MODERN GEOMETRY.
Then SM 2 - SN 2 = MV-NV = AW- AN" ;
SN 2 -SH = BN 2 -BL 2 ;
SL 2 -SM 2 = CI? -CM*;
.-. (Bit- CL 2 ) + {CM 2 - AM 2 ) + {AN 2 -BN 2 ) = 0.
Hence the perpendiculars from L, M, N on BC, CA, AB are
concurrent, say at R.
Triangles ABC, LMN which are thus related, are said to be
mutually Orthologic.
To determine the trigonometrical relation between S and B.
Since SB, SC are perpendicular to LN, LM,
.: LB8C-X = tt-L;
so that aa — II. sin (w— L) sin A/sin (ir—L—A), from (65)
or x = Il.sin LsinA/sin (L+ A).
Now let x', y', z' be the b.c. of B referred to LMN, and let
II ' be the power of R for the circle LMN.
Reasoning as before, we have
x 1 = Il'.sin {it— A) sini/bin {w—A — L)
= IT. sin A sin Z/sin {A + L) ;
.-. x/x' = y/y' = z/z' = area ABC/LMN
= n/n'.
Hence the b.c. of 8 with reference to ABC are as the b.c. of
R with reference to LMN; and the proportion is that of the
areas of the triangles, or of the powers of 8 and R for their
respective circumcircles.
83. Antipedal Triangles.
Let def be the pedal triangle of 8.
Then, since dS is perpendicular to BC, eS perpendicular to CA,
fS perpendicular to AB are concurrent at 8, the triangles ABC,
def are orthologic, so that the perpendiculars from A on ef, from
B onfd, from C on de, are concurrent — say at S'.
Through A, B, C draw perpendiculars to 8' A, S'B, S'C, form-
ing the triangle D'TS'F' — the Antipedal Triangle of S'.
The sides E'F 1 , ef (being each perpendicular to S'A) are
parallel ; and therefore def, the pedal triangle of S, and I/E'F',
the antipedal triangle of S', are nomothetic.
It follows that
D' = d = *, W-e = a, F'=f=v.
ANTIPEDAL TRIANGLES.
57
Also
BS'G = 7T-D 1 = tt-A, OS' A - a—/*, AS'B - «•- v.
Again, since 4B0 and def are orthologic, the b.c. of S' for
JBC are as the b.c. of S for def.
Therefore, taking aBy, a'B'y' as the n.c. of 8, S', respectively,
A. BS'G _ _ area of ABO _ A ,
A . e£/ ' area of def V '
.'. aa' = By sin A.A/U.
But, from (65),
n _ abc sin (i? + ;*) _ a6c sin ( + v) ,
M' sin/* ' y — ~M' sinv '
and
so that, finally, a'
But
A/Z7= M/2A;
abc sin(.B + ju,) sin (C + v)
M sin fi. sin v
a&c sin (.4 + A.) _
— ~~M' sin A. '
aa' = aVc2 siri (A+\) sin(B + M) sin(0+v)
M 2 ' sin A sin /a sin v
= BB' = yy', from symmetry.
Thus 8, S' are the foci of a conic inscribed in ABO.
(65)
In Germany these points are called " Gegenpunkte." In this
work the name " Counter Points " will be used.
58
MODERN GEOMETRY.
84. To determine the area (V) of JD'E'F', the antipedal
triangle of S, in terms of A, fi, v. (Gr.)
Henceforth the areas of the pedal triangles of 8, 8', &c, will
be denoted by U, U\ &c, antipedal triangles by V, V, &c.
Let V 2 , 3 be the circnmcentres of the circles S'BD'C,
S'CE'A, S'AF'B; S'D', S'F\ S'F' being diameters.
Then X A' = ^.acotD' = \.a cotX ;
.-. AO.BG = i.a 2 cot\.
Now 0, is the mid-point of S'D' ;
.-. area S'BD'C = 2 . BO.CS' = 2 (i . a? cot \+S'BC) ;
.-. F' = S'BD'C + S'CE'A+8'AF'B
— \ (a? cot \ + & 2 cot /a + c 2 cot v + 4A)
= *«"■
But
(65)
U" = 2A 2 /M".
.-. UV = A 2 ;
or, the area of ABC is a geometric mean between the area of
the pedal triangle of any point, and the homothetic antipedal
triangle.
This is a particular case of the more general theorem given
in (158).
ANTIPEDAL TRIANGLES. 59
85. To determine the n.c. (u, v, w) of T, the centre of similitude
of the homothetic triangles def and D'E'F'. (Q-.)
The ratio of corresponding lengths Td, TB' in def and B'E'F'
= VU : W = JUT' : V = A : V'?= JJ : A.
Therefore, since d, B' are homologous points,
u/B'D" = Td/m = Z7/(A-Z7).
Now the perpendicular from d' on /I0 = B'+a! cos 0,
„ ^17? = y' + a! cos JS;
.-. d'O = (|8' + a' cos C)/sin G, d'B = (■/ + a' cos £)/sin B ;
.-. B'B" = d'd" = d'B.d'O/d'S' ;
.-. lt - m . ^+ a ' OOBOf )(y f + B,ooBj ) i
a', sin B sin G
where m = P/(A — TT).
Example. — The orthocentric triangle, or pedal triangle of H,
is homothetic to T^T^ (Fig., p. 89), which is formed by
the tangents at A, B, G, and is therefore the antipedal triangle
of 0.
Here a' — B cos A, &c, and the formula gives
n D sin 2 A cos B cos G ■ A , ,
u — ZE. - — - <x sm A tan A.
1 — 52 cos A cos is cos G
Since 3 and are homologous points, being the in-centres of
the two triangles, the point T lies on OBL.
86. Let Si be the inverse of 8 for the circle ABG, as in (67),
and let (^e,/, be the pedal triangle of S v
Then, since 0^8^ e^, f^, perpendiculars to BG, GA, AB, are
concurrent at S lt therefore the triangles ABO and d^f^ are
orthologic.
It follows that the perpendiculars from A, B, G on e,/,, ^d v
d^ respectively meet at a point — call it £/.
Through A, B, G draw perpendiculars to 8-^'A, S^B, 6'j'O
forming the triangle B^E^F^, the antipedal triangle of S/.
The sides E^F( and e^/j, being each perpendicular to S-[A,
are parallel, so that d^/,, the pedal triangle of 8 V and D 1 ' J E 1 '2' , 1 ' )
the antipedal triangle of 8-[, are homothetic.
Hence D{ = d^ = X, E{ = e ; = n, F x ' — / L = v.
60
MODERN GEOMETRY.
We have now jour triangles, viz. : the pedal triangles of S, S„
and the antipedal triangles of S', $,', each of which has angles
Let a,/3iy, and a 1 ' / 8 1 ''fr' be the n.c. of ty and S/.
From (82) the b.c. of S/ in 45(7 are as the b.c. of S, in d^/,.
.-. ^^ = ... = ^ ; (Br, = area of ^/O
But, from (68),
so for yj ; and
Hence, finally,
aoj' = B^ . sin A.A/U V
8 = ^£ sin (5— /*)
Pl — Jlfi" sin/x ;
17, = 2A ! /Mi.
/ _ a&c sin (B—fi) sin (0— v) _
-M, sin /x sin v
, _ aW sin(A —\) sin (B—fj.) sin (C— v)
"l a l „»« • : — ; — : *
Mj sin A. sm t> sm v
— A A' — Yi^A from symmetry.
Hence 8 1 and S/ are a second pair of Counter Points, being
the foci of a conic touching the sides of ABO.
ANTIPEDAL TRIANGLES.
61
87. To determine the area (F/) of the antipedal triangle
Dt'EJFJ in terms of A, ju., v. .(G.)
Let 0/, 2 ', S ' be the circumcentres of the circles SiBD^C,
8,'GE^A, S.'AF.'B- S/D/, Si'-®/. S/J 1 / being diameters, and
0/, 0./, 0,' lying on A'O, B'O, G'O respectively.
Then 0,'A' = %a cot D/ = |a cot A ;
.-. A0 1 'BG = Ja 2 cotA.
Now 0/ is the mid-point of S/D/
Therefore, if (a/AV) be the n.e. of 5/, and ^ be the per-
pendicular from .D/ on BU,
fc. + a,' = 2.^0/;
... *,_«,' = 2 (A'C-O;
.-. a . b;bc-s;bg = 2 {o;bg-s;bg)-
= 2(|a 2 cot\-S 1 / B(7).
So JE^'CM + S/CM = 2 (±6 2 cot jn + S.T.-I ) ;
F;AB-S-?AB = 2(ic 2 cotv-S/Afl).
Adding, we have on the left side
B^BG + F 1 'AB-(ABO--E 1 'GA) ; [S'BG+S'AB-S'GA = ABU]
= D 1 'BG+F 1 'AB-(F 1 'BC+F 1 'AB)
= (D 1 'BG+F 1 , BG) + (F 1 'AB-E 1 -AB)
= e;bb;+e;bf; = b;e;f,' = v;.
62 MODERN GEOMETRY.
Hence Y( — \ (a 2 cot X + 6 s cot /x + c 2 cot v — 4A)
— AJlf
Now area (U,) of d&fr is 2^/M, ; (68)
.-. ^F/ = A 2 .
Note also that difference of antipedal triangles of 8', 8/
= v- y; = pr-f-M", = 4a.
The points 8' and 8^, having similar antipedal triangles, are
called " Twin Points."
Of the four points 8, 8', S lt $/ :
(a) 8 and &' x are Inverse Points, with similar Pedal triangles.
(6) S' and S/ are Twin Points, with similar Antipedal triangles.
(c) (SS') and (8^') are pairs of Counter Points, the Pedal
triangle of either point of a pair being homothetic to
the Antipedal triangle of its companion point.
CHAPTER VIII.
THE ORTHOGONAL PROJECTION OF A TRIANGLE.
88. * AL being a given axis in the plane of the triangle ABG,
it is required to determine the shape and size of the orthogonal
projection of ABG on a plane X, passing through AL and
inclined at an angle ti to the plane ABG.
Draw AM perpendicular to AL.
In AM take a point T, such that AT I AM = cos 0. Draw
BUb, CVc perpendicular to AL, cutting LT in U and V.
Then, since Ub/Bb = Vc/Gc = AT/ AM = cos 0,
A UV represents, in shape and size, the orthogonal projection of
ABO on the plane X.
* In writing Sections (88-90) I have drawn on Professor J. Neuberg's
"Projections et Contre-projections d'un Triangle fixe," with the author's
permission. To avoid overloading the chapter, the Counter-projection
theorems are omitted.
63
64 MODERN GEOMETRY.
On either side of the common base BG a series of triangles
is described similar to the orthogonal projections of ABG on a
series of planes X passing through the common axis AL.
It is required to determine the locus of the vertices of these
triangles.
Draw AtS perpendicular to LT, cutting the circle ALA^M in
8. The triangle SBC will be similar to A JJV, and therefore to
the projection of ABG on the plane X.
Draw SNS' perpendicular to BC.
Then Z ALT = tAT or SAM
= SLM.
And A, t, S, N are right angles.
Therefore the figures ALtT, SLNM are similar.
And LU : LB = LV : LO = LT : LM.
Therefore the figures ALVVtT and SLBGNM are similar.
Therefore the triangle SBC is similar to AW.
■ Hence the vertices of all triangles SBG, described on BG, and
having SBG = AUV, SCB = AVU, BSG = UAV,
lie on the circle ALA V
The angles of projection range from
B = to 8 = fyr.
When 6 = 0, cos 6 = 1, so that T coincides with M, S with
A v and 8' with A.
As increases from to \ir, T travels from M to A, S from
A 1 to M, and 8' from A to M.
Hence the locus of S is the arc AMA t on the side of A A'
remote from the axis AL.
89. When a series of variable axes AL V AL 2 , ... are taken, the
point A t , being the image of A in BO, remains unchanged.
Hence each position of the axis AL gives rise to a circle
ALA X M passing through the two fixed points A and A lt so that
this family of circles is coaxal.
Let l)A or BA l = k, and let DB = h, where B is the centre
of the circle ALA X .
Then, with B as origin and BA as j/-axis, the equation of the
circle is x 2 +y i —k 2 = 2hx,
h being the variable of the coaxal system.
We may now deal with the problem : To determine the
plane X on which the orthogonal projection of ABG has given
angles X, /x, v.
ORTHOGONAL PROJECTION OF A TRIANGLE.
65
Construct the triangle SBC, so that SBC = /x, SCB = v, and
hence BSC = \.
Let the circle A>IA 1 cut BO in L and M.
Then, if AL and S are on opposite sides of AA lt AL is the
required axis.
Draw LtT perpendicular to SA.
Then the required inclination 6 of the plane X to the plane
ABC is given by
cos 6 = AT I AM.
90. The triangle ABC being projected on a series of planes
making a constant angle a with the plane of ABC, and the
triangles SBC being drawn, as before, similar to the successive
projections, it is required to determine the locus of S.
Draw AL parallel to the line of intersection of the planes ;
then the original projection is equal and similar to the pro-
jection on a parallel plane through AL.
Determine S as before, taking T such that
AT/AM = cos a.
Join SL, SM, cutting AA 1 in R and H'.
Then S is the orthocentre of LH'M, so that
BH.BE' = DL.DM=W.
66 MODERN GEOMETRY.
AT tun ALT
Again,
AM tsmALM
tan SUf HD
' tan/l^.l/" A,D'
.'. HD = h cos a = constant,
H'D = h sec a = constant,
so that B" and H' are fixed points.
Hence, since HSH' is a right angle, the point >S describes a
circle on ME' as diameter.
The equation to the circle HSH' is
» 2 + (y—Tc cos a) (y— k sec a) = 0.
For another series of planes inclined at a constant angle a',
the points H and H' would be changed to E x and H,', where
DH t = & cos a', Dfl",' = & sec a',
and BH l ,Dn i ' = k 1 .
The series of circles are therefore coaxal, having A and A t
for limiting points, and therefore cutting orthogonally the
former series of circles, which pass through A and A v
91. A triangle ABO, with sides a, b, e and area A, is projected
orthogonally into a triangle A'B'C, with sides a', V, c! , angles
X, /u., i', and area A' ; the angle of projection being 0, so that
A' = A cos 6.
To prove 2.ft' J cot A — 2A(l+cos 2 0) ;
2. a 2 cot A = 2A (sec 6 + cos $).
Let hu A 2 , &, be the heights of the points A, B, G above the
plane A'B'C. '
Then 1,,—h. = vV-
.-. v/V_ a '*± v/6 2 -6' ! ± v/r-c' 2 = 0,
the signs of the surds depending on the relative heights of
A, B, G.
This leads to
46'V 2 -(6' 2 + c' i -a' 2 ) 2 +4.bV-(6 !, +c s -a 2 ) ! or 16A J +.16A' 2
= 22.a' 2 (6 2 + c 2 -a 2 ) = 22 .a 2 (&' 2 + c' 2 -a' 2 ).
Now 6 2 + c 2 — a 2 = 4A.cot J 4 ;
b' 2 + c' 2 - a' 2 = 4A'. cot A. = 4A . cos 6 cot A.
Hence 2 . a' z cot A = 2A (1 + cos 2 6) ;
2. a 2 cot A. =2A(sec0 + cos0).
ORTHOGONAL PROJECTION OF A TRIANGLE. 67
The Equilateral Triangle and the Brocard Angle.
When ABG is equilateral, we have
(o'» + 6'« +c '«).l/ V '3 = 2A(l + cos s «) = 2A'(sec0 + cos0).
But, if cu' be the Brocard Angle of A'B'O,
oot <*>' = (a" +■ 6' 2 + c' 2 )/4A' ; (131)
.-. cot co' = i/3/2 . (cos (9 + sec 6) .
The Brocard Angle therefore depends solely on the angle (6)
of projection. It follows that all equilateral coplanar triangles
project into triangles having the same Brocard Angle.
92. Antipedal Triangles and Projection. (G-.)
Let I'm'n', Ifmjnf be the sides ; V, F/ the areas ; and A, p, v
the angles of B'E'F', D,'E 1 'F 1 ', the antipedal triangles of 8', «,'.
From (84),
2V = 2.a 2 cotA+4A = 2A(sec0+cos<9)+4A;
.-. V'/A = ( y^cl»+ v/c^sl) 2 .
so f//a = ( yiicT- y^ey ;
... VV'-^V,' = 2. \/Acos0 = 2.i/A';
and i/F'+i/F' 1 ' = 2.v/AlecT=2.v/A 1 ;
where Aj is the area of the counter-projection of ABO: that is,
the triangle whose projection (for 6) is ABU.
Now the triangles V', V-[, with the projection and counter-
projection, are all similar, having angles A., //., v ; so that their
corresponding sides are as the square roots of their areas ;
.-. ?-l 1 ' = 2.a'; Z' + Z 1 ' = 2.o I .
Hence, if two antipedal triangles be drawn having the same
angles as the projection, the sides of the projection are half the
difference of the corresponding sides of the antipedal triangles.
This theorem, of which the above is a new proof, is due to
Lhuillier and Neuberg.
93. The triangle ABG is projected orthogonally on to a plane
inclined at an angle to the plane of ABG.
If U, the line of intersection of the planes, make direction
apgles u v k», u s with the sides of ABC, it is required to deter-
mine the lengths a', V, d of the sides of the projection A'B'G' in
terms of m„ m 2 , u s , and 6.
Let cos 8 = h.
68
MODERN GEOMETRY.
Draw perpendiculars Bb, Cc to the line V, and take
bB' = k.bB, cC = k.cG,
80 that B'C (= a') is equal to the projection of BG.
Now the projection of B'C on Bb
= B'b-G'c = k(Bb-Gc) = k.asinu,;
and the projection of B'G' on U = a cos w, ;
.-. a n = B'G n = a 2 (cos 2 «, + fc 2 sin 2 «,)
= a 2 {l-(l-fc 2 )sin 2 Ml ]
= a 2 (1— sin 2 ^ sin 2 6).
The dimensions of the projection depending, of course, only
on the direction, not on the position, of U.
94. Pedal Triangles and Projection. (G-.)
To prove that, as the plane of projection revolves round U,
the projections of ABG are similar to the pedal triangles of
points lying on the circumdiameter TOT', where T is the pole
of the Simson Line parallel to Z7.
Let XYZ, X'Y'Z' be the Simson Lines of T, T'-
Then XYZ, being parallel to U, makes angles «,, «,, w 8 with
the sides of ABC, and therefore
IOTA or OAT= u v (37)
ORTHOGONAL PROJECTION" OF A TRIANGLE. 69
In TOT' take any point P, and let OP = k'.B.
Let u, v, w be the sides of the pedal triangle of P.
Then, in the triangle OAF,
AP* = E ! + fc' 2 E 2 -2fc'JB 2 cosO— 2 Ml )
So that
= ff (l + ^l__^_ sin2ttl J
4^ = i.AP 1 sin* A = a?(l + h'y \ 1- 4fc ' sin 2 Ml \ .
But a' 2 = a 2 {l-(l-^)sin 2 « 1 }. (93)
Hence ^ 2 = (iw
and a' : h' : c' = u : u : i«
(so that the projection on the plane 6 is similar to the pedal
triangle of P), provided that
&k'~
(i + k'y
1-F,
f 1 — h 7/ 1 — 1c
k = i+k" y = irr
so that P, and its inverse point P', are given by
TP/T'P = TP'/T'P' = cos 6.
From (36) the point T is found by drawing chord At parallel
to U, and the chord tXT perpendicular to BO.
95. Next, let the plane of projection be inclined at a constant
angle u. to the plane of ABO. (Gr.)
1 — k __ 1 — cos a
1 + k 1 + cos a
.-. OP = B tan 2 £a, OP' = B cot 2 £a.
If, therefore, circles be drawn with centre and radii
tan 2 |a.JS and cot 2 %a.R, any point P on one circle, and its
inverse point P' on the other circle, will have their pedal
triangles similar to the projections of ABC.
Another Proof. — Let ABO be projected on the plane a into
A'B'C, whose angles are X, fi, v. Let P (or its inverse P') be
a point whose pedal triangle has angles A., ft, v.
T ., . ,; i — k x — cos a , 2 ,
In this case, k = n i ; = ., = tan' ^a ;
70 MODERN GEOMETRY.
tiiot, nvm- — a s cot a + 6 a cot ^ + c' cot v—a a , B , .
1 hen W /i« - a2 cQt k + htoobll + & cot „ + 4A tM-)
_ 2A (sec a + cos a) — 4A ^_ (1— cos a) 2
— 2A(seca + cosa)+4A — (1 + cosa) 2 '
.-. OP/-R = tan 2 ia; so OP'/B = cot 2 £a,
as before.
96. A triangle XY.Z is projected orthogonally on a series of
planes inclined to its own plane at a given angle 6. A point is
taken such that its pedal triangle with respect to ABC is
similar to one of these projections.
To determine the locus of the point.
Draw inner arcs BQC, CQA, A QB containing angles A + X,
B+Y, G+Z: a point Q being thus determined whose pedal
triangle has angles X, Y, Z.
Let AQ, BQ, GQ (or AQ', BQ f , CQ') meet the circle ABC
again in seyz (or x'y'z') : then it is lmown that
the angle x or x' = X, y or y' = Y, z or z' — Z. (56)
Take Q as inversion centre, and let
(inversion-radius) 2 or h % = power of Q for ABC
= R 2 -OQ* = OQ.OQ'-OQ 2
= QO.QQ'i
so that in this system Q' is inverse to 0.
Also V = AQ.Qx = BQ.Qy = CQ.Qz;
so that xyz inverts into ABC,
97. Lemma.
If with any centre K and any radius p, any four points
I), E, F t G are inverted into D', E', F', G', then the pedal
triangle of G with respect to BFF is similar to the pedal
triangle of G' with respect to D'E'F'. For
B'G'/BG = KD'/KG, and E'F'/EF = EE'/KF;
.-. (B'G'.E'F)/(BG.EF) = (KD'.KE')/(KG.EF)
= (KB'.KE',KF')l(p\KG)
= (E'G'.F'B')/(EG.FB)
= (F'G'.D'E')/(FG.DE),
by symmetry, so that the theorem is proved.
ORTHOGONAL PROJECTION OF A TRIANGLE.
71
In Section (95) substitute the triangle xyz for ABC. Then
the circles with common centre and radii Bfi, B/f, are the
loci of points whose pedal triangles with respect to xyz are
similar to the projections of XYZ on planes inclined to the
plane of XYZ at an angle 6.
Now invert, with centre Q, radius k. Then xyz inverts into
A BO, into Q', and the two concentric circles (0, Bt 2 ) and
(0, B/t 2 ) into circles of the coaxal system which has Q. and Q'
for its limiting points.
Let 2? be a point on either concentric circle, inverting into 8,
a point on one or other of the coaxal circles. Then, by the
Lemma, the pedal triangle of S with respect to ABC is similar
to the pedal triangle of B with respect to xyz ; that is, to one of
the projections of XYZ.
Hence the required locus consists of these two coaxal circles.
98. Let the circumdiameter OQQ' be cut by the concentric
circles in LL', MM' : and by the coaxal circles in IV, mm', : so
that L and I, ..., are inverse points in the (Q, Jc) system.
The centres co, to' and the radii p, p' of the circles W, mm' will
now be determined in terms of X, Y, Z, t,
Ql = WjQL = ky(0Q-Bt 2 ), QV = k 2 /(OQ + Bt 2 ) ;
.-. 01 = 0Q+ Ql = [BiB-OQ.t^/iOQ-Bf),
01' = [R(B+OQ.t 2 )]KOQ + Rt 2 ), [k 2 = B 2 -OQ 2 ];
P /R = ±(0l- 01') /B = kH 2 /( 0Q a -E¥),
0u>/0Q = %(0l+0l')/0Q = [B'l'l -**)]/( OQf-BH*).
72 MODERN GEOMETRY.
And, writing 1/i 2 for t'\
0m= [_R(OQ-RP)~\HB-OQ.P),
Om' = [R(Rt 2 +OQ)]/(R + OQj),
p'/R = h 2 f/(OQ\t i -R 2 ),
(ht/OQ = [B , (l-* 4 )]/(#-OQ , .f).
Hence 01 . Om = E 2 = 01' . Om',
so that the circles IV, mm', which are inverse to LL', MM' in the
(Q, K) system, are mutually inverse (as are LL', MM') in the
(0, R) system. Since the pedal triangle of Q has angles
X, Y, Z, it follows, from Section (64), that
Ony-D* — "' cot X + V cot Y + c " cot g ~ 4A
W/ _ a 2 cotX+6 2 cotr+c 2 cot^+4A'
so that now p, p', Ota, Oo>' are expressed in terms of X, Y, Z, t.
When OQ = Bfi, or when Q lies on the inner concentric circle
LL', then 0u> and p are infinite.
Hence the (Q, k) inverse of the circle (0, OQ) is D.E7D',_the
Radical Axis of the coaxal system, bisecting QQ' at right
angles.
And if N be the pole of BED' for the circle ABC, it is easily
proved that the (Q, k) inverse of the circle (0, OQ') is the
circle ON.
99. A case of great beauty and interest presents itself when the
triangle XYZ is equilateral. For then Q, QJ, having equilateral
pedal triangles, are the Isodynamic points 8, 8„ lying on OK,
the Radical Axis of the coaxal system is now the Lemoine axis ;
and N, the pole of the Lemoine axis for the circle ABC, is the
Lemoine point K.
It follows that in the (8, h) system, the Lempine axis is the
inverse of the circle (0, OB) ; while the Brocard circle is the
inverse of (0, 08j).
Suppose the equilateral triangle XYZ to be projected into a
triangle with angles Xp.v, sides Imn, area A', and Brocard angle <p.
Then, from Section (91),
cot<£ = (Z 2 +m 2 +ra n -)/4A' = -|.v/3.(sec0 + cos0)
= ■y3.(l + t i )/(l-t i ); [£=Etan§0]
.-. t* = (cot <f>— J3)/cab <p+ V3).
Therefore <p is constant, as 6 is constant.
Hence the remarkable property of this coaxal system of
Schoute circles, viz., the pedal triangles of all points on the
circumference of any one circle have the same Brocard angle.
ORTHOGONAL PROJECTION OF A TRIANGLE. . 73
100. To prove that the Brocard angle <j> is equal to the acute
angle (less than 30°) between wO and OU. (G.)
From Section (98),
O&VB 1 = (a 2 cot 60°+ ... -4A)/0 2 cot 60° + ... + 4A)
= (cotw— v/3)/(cot<u+ a/3).
And t l = (cot <j>— v/3)/(cot <f> + ^3) (Section 99).
Therefore, from Section (98),
0<a/08 = [E 2 (l-i 4 )]/(OS 2 -EV)
= (coto>+ v'3)/(cot &)— cot<£).
In our calculations, Section (98), we took Rt 2 < OQ, so that
<o lies to the right of Q,, and therefore of QJ (here S^).
Let 0Q<o = 180— x-
It is known that
on& = 30°, ons, = 150° ; (160/.)
.-. o«>/08= 0u/ on. on/ os
= sin x/sin ( x — u>) .sin (*> + 30°)/sin 30°
= (cot id + \/3)/(cot<i) — cot x );
■'• 4> = X-
Many years ago Prof. Dr. P. H. Schoute proved that the locus
of a point whose pedal triangle had a constant Brocard angle is
a circle of the coaxal system whose limiting points were 6, 8 V
His proof, I believe, was analytical.
* Read sections 99, 100 after Chapter XI.
CHAPTER IX.
COUNTER. POINTS.
101. We now proceed to a further examination of the two
pairs of Counter Points (S, 8') and (S x , S/), where the pedal
triangle of S is homothetic to the antipedal triangle of 8', while
the pedal triangle of S' is homothetic to the antipedal triangle
of S, and so for S t and $/.
Let a/fy, a'/3y be the n.c. of 8, S'.
It has been shown that
a < = ap — yJ — aV ^ ain (A + k) Bin (B + i>.) sin (C + v) ^ / g ^
' if 2 ' sin A. sin yu. sin v
74
COUNTER POINTS. 75
It follows that 8 and S' are the foci of a conic inscribed in
ABO ; that the two pedal triangles def, d'e'f have the same
circumcircle. the Auxiliary Circle of the conic ; that the major
axis = 2p, the diameter of this circle, the centre lying midway
between S and S', while the semi-minor axis q is given by
q 2 = aa', &c.
The points 8, 8' are often called 'Isogonal Conjugates,"
because, by a well known property of conies,
Z SOB = S'CA, 8BA = S'BC, 8 A C = S'AB,
the pairs (8A, S'A), &c, being equally inclined to the corre-
sponding sides.
The line S'A is then said to be isogonal to SA, S'B to SB,
S'G to SO; so that 8', the counter point of S, lies on any line
isogonal to SA or SB or SO. ~ '
Let aBy be any point L on BO, and a'B'y' its counter point.
Then BB' = y/ = aa' = 0. (a = 0)
But B, y are not zero ;
.-. B' = y' = 0;
i.e. the counter point of L is A.
If S be on the circle ABC, draw chord ST perpendicular to
BO, and draw diameters SS, and TT,.
Obviously -AT, is isogonal to AS; also AT t is parallel to the
Simson Line of S lt and therefore perpendicular to the Simson
Line of 8.
Hence the counter point of S is at infinity, being common
to the three parallels AT V BT„, OT 3 which are perpendicular
to the Simson Line of S.
102. To determine the relations between A., //., v, the angles
of def, and A.', fi!, v', the angles of d'e'f.
It is known that tBSC = A + \; so B8'C = A+X'. (56)
But BS'C = tt-BB'O = 7T-A ; so B8G = v-k'.
.: B8G + BS'0 = (A+k)+(-n—k) = Tr + A.
And A+k = BSC = ir-k';
.'. k+k' = 7T-A.
.'. sink' = sin. (A + k), sin A. = sin (A + k') ;
76
MODERN
GEOMETRY.
also
abc
sin V
sin A.
_, _ ahc sin// sinv'
M" ' sin fn sin v
(83)
, » , , <x 2 &V t o sin X' sin u.' sin v'
and <r = aa = m' . , where m = — : — - — ;— £- — : .
M' em A sin /*. sin v
Let EL' be the power of S' ; V the area of d'e'f ;
2tf' = a 3 cotX'+...+4A.
Then JIM = 2E.a6c = WW. (64)
And 2p 2 .sinX sin/u sinv = Z7 =2A 2 /if; (65)
2p 2 . sin A' sin / sin v' = 17' = 2A 2 /Jf '.
.-. w 2 = c"/Z7 = jf/jtf' = n'/n ;
.-. U' = m t U = m\2**/M,
and Jf ' = M/m 2 , n' = to 2 II.
q. tt sin(.4+X) sin A ~ sin X' sin A
Since aa = n. ^ — ; — '- = n . : ;
sin X sm X
and, similarly' ,
/ _ rr' sin(^i + X') sin A _ „, sin X sin A
aa — ii . ; — — ii . : — — — .
sin X sin X
.-. 4i?V = 4E 1! .aa' = nn';
... (#2 _ OS 2 ) (E 2 - OS' 2 ) = 4BV ;
a result due to Professor Genese.
103. The equation to the minor axis of the conic which has
8, 8' for foci. (H. M. Taylor)
Let 7!"!, ir it ir 3 be the perpendiculars on the minor axis from
A, B, C, so that the equation is
7Tja . a + 7r 2 b . /3 + ir s c . y = 0.
A diagram shows that
8A 2 -S'A 2 = 2.S8'.7r 1 ;
8 A sic A = ef = 2p sin X ; (57)
- . ^.sin'A x (sin 2 X— sin 2 X') a sin(X— X') sin^l.
[X + X' = 7I— 4]
Hence the equation to the minor axis is
a.sin(X— X')+/3. sin (/■ — //) + y. sin (v— v') — 0.
The proof here given is by the present writer.
COUNTER POINTS.
77
104. Lemma. — The tangent be to the conic being drawn
parallel to BG, to prove
SA.S'A -AG.Ab= AB.Ac~
Let a, 8, y, 8 be the angles subtended at S by the tangents
from B, G, c, b.
Then ASb = ASp-bSp = %(2y + 2t)-8 = y;
and, regarding the parallel tangents as drawn from a point D
at infinity, so that SD is parallel to BO,
/.BSl = DSn = i(2/J + 2y) = B + y;
.: DSG = DSl-CSl=(B+y)-B = y;
.-. S'CA = SGI - BSG = y = ASb.
Also S'A C = SAL.
Hence the triangles ASb, ACS' are similar;
.-. AS :Ab = AC : AS' ;
.-. SA.S'A = AG. Ab=AB. Ac.
The theorem is due to Mr. B. P. Rouse, the demonstration to
Mr. R. F. Davis.
78 MODERN GEOMETRY.
105. The coordinates of the centre <r of the conic. (Gr.)
Let a oj 8 y , a '/3 'y ' be the n.c. of o- , the centre of the conic,
referred to ABC and to the mid-point triangle A'B'C respect-
ively.
Let h x be the perpendicular from A on BG.
Then a +■ a ' = ^\ ;
.-. 4ft.o ' = 2B(h,— 2a ) = 2Bxperp. from A on be
= 2Rx.Ab sin B or 2Bx Ac sin C
= AG.Ab or AB.Ac
= &'A.S'A (from Rouse's Theorem).
But 8 A . sin 4 = e/ = 2p . sin A ; (57)
.•. 4E. a ' = 4p 3 . sin A sin A'/sin s 4. ;
.•, aa ' = 2j0 9 . sin A. sin A'/sin 4,
giving the absolute A'B'C' b.c. of o- ; and since
eM ' + &A,'-|-cy ' = A,
,-. p 1 — JA/JV, where JV = 2 . sin X sin A'/sin .4.
106. The conic touching BO at Z, to prove
Bl : Gl = sin ju sin /x'/sin _B : sin v sin v'/sin (7.
Project the conic (ellipse) into a circle of radius q, the angle
of projection being 6 = cos -1 q/p.
Let the centre o- be projected into cr 1 , and ABG into A-fi-^C^.
Then ^..B 1 a- 1 G 1 = A.Bcr C xq/p ; .'. q.a-^ = aa Xq/p ;
fflj cc tta ;
.-. Bl: Gl- B& : Cy, = s l —b 1 : s-c,
= aa — 6/3 +cy : aa +bj3 -cy„
= W : ey ' (14)
= sin /n sin ///sin B : sin v sin v'/sin 0.
The theorem is by Mr. H. 1L Taylor; the proof is by the
present writer.
COUNTER POINTS.
79
107. Let to, m be the orthopoles of the circumdiameters passing
through 8 and 8' ; i.e. the points on the Nine-Point circle
whose Nine-Point circle Simson' Lines are parallel to the
diameters OS and 08' (49).
The pedal circle of every point on the diameter through S
passes through. <■> ; therefore def, the pedal circle of 8, passes
through to; similarly the circle d'e'f passes through to'. (78)
Therefore def d'e'f, the common pedal circle of S and 8', passes
through to and <u'.
And therefore toto' is the Radical Axis of the two circles. (G-.)
^--ir:::-
When OS' falls on 08, to' coincides with to, and the pedal
circle touches the Nine-Point circle at to.
108. Aiyar's Theorem.
If 0' be the Nine-Point centre
Then
OS. 08' = 2E.0'o-„.
Let 0,0,0, and 0,'0 2 '0,' be the direction angles of OS, OS'.
The power of A' for the pedal circle = A'd.A'd'
= 08 cos O x . OS' cos 0/.
But since too/ is the Radical Axis of the two circles, the
power of A' for pedal circle
= perpendicular from A' on coco' x 2 . OV .
And this perpendicular from A'
= A 'to . .4 V/E = E cos 0, . E cos 0/ ; (44)
.-. 08.08' ='2R.a<r,.
(V. Ramaswami Aiyar.)
80 MODERN GEOMETRY.
Note that the A'B'C equation to wa>' is
cos8 1 cos0- l ',aa'+ ... = 0:
for, from (50), co and <o' have n.c. (sec0 x ... sec 9/...).
109. M'Cay's Cubic. For this occasion take (Imn) as the
n.c. ABC coordinate? of S ; so that (1/Z, 1/m, 1/w) are the n.c.
of «'.
The equation to SS' then is
Z(m 2 -m 2 )a+... =0.
When SS' passes through O, we have
IQnz 1 — n-) cos A + ... =0.
So that -8 and S' lie on M'Cay's Cubic,
a (/J 1 — y*)cosA+....
And conversely, if any diameter TOT cut this cubic at S, S'
(the third point is O), then S, S' are counter points, and their
pedal circle touches the Nine-Point circle at u>, the orthopole
of TOT'.
We have already met with this circle in Section (45) : its
centre is 0„ O'O^ is a straight line, so that now Aiyar's
Theorem may be written OS. OS' = 2B(JB— p), and if the
circle cut TSS'OO^' in fc, &', then <ak, wh' are the Simson Lines
of T, T.
110. Counterpoint Conies. (C".)
The point P moving along a given line TT', it is required
to determine the locus of its counter point Q.
Let TT cut the sides of ABC in L, M, N.
Then the Q locus passes through A, B, C the counter point,
of L, M, N. (101)
Draw the chord Aa parallel to TT', and at parallel to BC.
The counter point of t (101) is the point at ao on Aa, and
therefore on TT'. But this point being on TT', its counter point
must be on the locus of Q.
Hence t is a point where the Q locus cuts the circle ABC ;
also (101) t is the pole of the Simson Line perpendicular to TT.
Denote perpendiculars from Q, the counter point of P, on
AC, AB, Bt, Ct by p, y, ft, y'.
COUNTER POINTS. 81
Then i- = sin C UG _ sinPAB .
y sin Q.I li sin PA. 0'
£ _ QB.nmQBt _ sin QGB sin QBt _ am PGA sin QBt
y' QO .sin Qvt sin QBO ' sin QCt ninPBA ' sin QCt'
JPJOL
1 ^^1
L
-J,
I / /
1 / / '
1 / / /
1 / / /
1 / / '
V / 4
oCk7
S*>
T\
yx
Let BP, OP cut Aa in m and «.
Then QBt = QBO+GBt = PPA + BC<x = PBA + BAm
= Brno. = BPT.
So QCt = QCB+BCt = PGA + GBo. = POA + nAG
= Pna = OPT.
§£_PA RB sin PPT _ PP. sin PPT _ q
" yy' PC • PA ■ sin C'PT PV.&inCPT r '
where p, q, r are perpendiculars from A, B, G on TT' .
Hence the locus is a conic passing through A, B, G, t.
If the sides of the quadrilateral were BA, BG, tA, tC, we
should have yy'/ aa ' = r /P»
111. To determine the directions of the asymptotes and axes.
Draw Tx, T'x' parallel to BG.
The points at infinity of the Q conic are the counter points of
T, T' ; and therefore lie on Ax, Ax', which are isogonal to AT,
AT', so that Ax, Ax 1 are parallel to the asymptotes.
But the arc tx = aT = AT'.
Hence tT, tT' are parallel to Ax', Ax, and therefore to the
asymptotes.
82
MODERN GEOMETRY.
112. To determine the Counter Point Conic of a circum-
diameter TOT'-
From (111), the Asymptotes are at right angles, since they
are parallel to tT, tT' ; therefore the conic is [a Rectangular
Hyperbola.
If TOT' be lu + mP + ny=Q or p.aa+ ... = 0,
the conic is l/a+ ... = 0.
To find the centre, we have
my + m/3 oc a, &c. ;
.". a <x I (— al + lmi + cn).
But I a: op, and, from (8), ap/B = 6 cos s + c cos 6 2 ;
whence u. on p cos 0„ &o.
Hence, from (50), the centre is u>, the Orthopole of TOT'.
113. To determine the Asymptotes. (Gr.)
Let u,X, wX' be the Simson Lines of T, T'.
From Section (41), if w„ t„ w„ h x are the perpendiculars from
A, B, G, H on oiXj ; and u. 2 , v 2 , to 2 , ft 2 those on <i)X 2 ; then
«, = 2B . cos <r l sin o- 2 sin o- 3 ,
u 2 = 2ff.cos (^tt — cr t )...
= 2.R . sin o"! cos cr 2 cos <r s ;
COUNTEE POINTS. 83
.". 2 . m,« 2 = JB 2 „sin 2o-, sin 2<r 2 sin 2cr 3
= 2 . i>ii)„ = 2 . w,iv„.
Also A t = perp. from H on ul,
= perp. from T on coXj
= 2B . COS (Tj cos cr 2 cos <r 3 .
So h, — 2B.cos (^77—0-,)...
= 2 J? . sin o-j sin <r, sin o- 3 ;
2.«jM 2 = ... 2h 1 h.,.
Hence wZ,, <oJ, are the Asymptotes of the Rectangular Hy-
perbola and the square of the semi-axis
= E?. sin 2o"i sin 2<r 2 sin 2cr s .
Since OTA or OAT = o-, ; (37)
.-. p = B. sin AOT = E.sin2<r.
Hence the square of semi-axis = pqr/B.
114. Produce Hid to cut the circle ABC in t'.
Then, since in lies on the Nine-Point Circle,
Ma = u>t'.
But H is on the Rectangular Hyperbola, and <o is the centre.
Therefore i lies on the Rectangular Hyperbola.
Therefore if coincides with t, the fourth point where the
Rectangular Hyperbola cuts the circle ABO.
Let <Xj, B v y 1 be the n.c. of S,, inverse to 8; and a/, /}/, y/
the n.c. of #,', the Twin Point of S'.
It has been shown that
_ abc sin {A — X) . , _ abc sin (2? — /a) sin ((7 — v)
1 JkT ' sin A. 1 M sin //, sin v
(68) and (86)
•■■ «,«/ = AA' = yi7,'-
So that /S/ is the counter point of S„ and therefore lies on the
Rectangular Hyperbola, which is the counter point conic of
TOT'
84
MODERN GEOMETRY.
115. To prove that a> is tke mid-point of S'S/.
We have shown that the pedal circles of all points on TOT'
pass through the orthopole <o.
Therefore the pedal circles of (88') and (S&') pass through o>.
Also, from Section (69), <o is the Centre of Similitude of def,
the pedal circle of S, and of d,e,/„ the pedal circle of $,.
Hence, if p, p l are the circumradii of these circles,
u>d tod x
P ~ Pi '
d, d' being homologous points in the similar triangles def, e^e,/,.
Draw o>k perpendicular to BO.
Then, since u> is on the circle defd'e'f,
tod . lad'
iak =
2p
and similarly, <ak = wd, . o>d 1 '/2p l ;
. tod' = tod/ ;
so that the projection of to on BO is midway between the pro-
jections of S' and S/ on BO.
So for OA, AB.
Therefore <o is the mid-point of S' and S/.
Hence, as the inverse points 8 and Si travel along TOT' in
contrary directions from T, their counter points 8' and S/ travel
along the Rectangular Hyperbola which passes through A, B,
and has <o for its centre. 8', S/ are always at the extremities
of a diameter of the Rectangular Hyperbola, and the difference
between the areas of their antipedal triangles is always 4A.
COUNTER POINTS. 85
116. Let V, m\ ri be the images of S' in BC, CA, AB. (G.)
Then, because <r is the centre of the pedal circle of (SS'),
.: <r S = <r S'; .: 8V = 2. <r i' = 2p.
Therefore a circle, centre 8, radius 2p, passes through I'm'n'.
Again, since S'a- = cr i8 and 8'<a = wiS/,
.-. SS 1 ' = 2.<r„o» = 2p,
since the circle def passes through <o.
Hence a circle described with centre S and radius 2p passes
through I'm'n' and 8^.
CHAPTER X.
LEMOINE GEOMETRY.
117. The Lernoine Point. — On the sides of the triangle ABC
construct squares externally, and complete the diagram as
given, the three outer sides of the squares meeting in A„ B„ 0,.
The perpendiculars from A, on AG, AB being b and equal to
c, the equation to AA, is /S/6 = y/c.
Hence AA V BB„ GO, meet at the point whose n.c. are
(a, 6, c),and whose b.c. are therefore (a 2 , V, c 2 ).
This point is called the Lernoine or Grebe or Symmedian
Point and will be denoted by K.
AK, BK, OK are called the Symmedians of A, B, 0.
The absolute values of the n.c. are given by u = ka, &c,
where 1c = —. — —. s .
d'+b'+c*
Produce AK to meet BG in K, ; then,
BK, : OK, = AAKB : AAKG
= ratio of b.c. z and y
= c 2 : ¥ ;
so that the segments BK„ OK, are as the squares of adjacent
sides.'
LEMOINE GEOMETRY. 87
118. K is the centroid of its pedal triangle def.
For AeKf=%.Ke.Kf sin A
oc be sin A ;
.-. A eEf = AfKd = A dKe.
So that K is the centroid of def.
Since the n.c. of are as (cos .4, cosB, cos 0), while
those of K are as (sin A, sinB, sin G), therefore the equation to
OK is sin(B-C).a+Bin(C-A).l3 + sin(A-B).y,
or (6 2 -e ! )/a 2 . x+... = 0.
The Power of K for the circle ABO.
Using the form of Section (60), we have
w-ok> = n = /w+- = , I 3a ' 6 ' ' .
(SJ + 2/ + Z) 2 (a 2 +6 2 + c 2 ) 2
119. If a, /3, y are the n.c. of any point, then a 2 + /3 2 +y 2 is a
minimum at K.
Since (a 2 +6 2 + c 2 )(a a + /3 2 +y 2 )
= (aa+6 / 8+cy) 2 +(6y-c / 8) 2 + (ca-ay)»+(a)8-6a)»
= 4A 2 +...;
therefore (a 5 +6 2 + c 2 )(a 2 +/3 2 +y 2 ) is always greater than 4A 2
except when a/a = /3/6 — y/c
or when the point coincides with K.
In this case, therefore, a 2 +/3 2 +y 2 has its minimum value,
4A 2
which is
a 2 + 6 2 + c 2
The sides of the pedal triangle of a point S are u, v, tr.
To show that m'+o' + ic 1 is a minimum when S coincides with
the Lemoine Point K.
From (57), u = r, . sin A ; (SA = r,)
.• . 4E 2 (w 2 + tr + w*) = a V + 6 V + c\\
But, since a 2 , b 2 , c 2 are the h.c. of K, we have, from (19),
a\ r,» + 6 2 - r 2 2 + c 2 . r 8 2 = (a 2 + 6= + c 2 ) (Jf S 2 - K0' + JS 3 ) ;
and the right-hand expression is a minimum when KS = 0, or
when S coincides with K.
The minimum value of uf+v^+w 1
o V+... _ a 2 +& 2 + c 3 „ _ o A +OT1
4A * (118) and (131)
88 MODERN GEOMETRY.
120. Let figures Yand Z, directly similar, be described exter-
nally on AG, AB, so that A in Y is homologous to B in Z, and
G in Y to A in Z.
Then, if S be the double point of T"and Z, the triangles SAC,
SBA are, similar, having
L SBA = SAG ; SAB = 8GA ; ASB = ASO,
so that S is the focus of a parabola known as Artzt's First
Parabola, touching AB, AG at B, G.
Let j9 2 , p a be the perpendiculars from S on AG, AB.
Then, from similar triangles SBA, SAG,
p.,:p s = AO:AB,
so that S lies on the -A-Symmedian.
Again, L BSK^ = SAB + SBA
= 8AB+SAG
= A;
.: BSG = 2 . BSK, = 2A = BOG;
so that S lies on the circle BOG.
Also, OSE 1 = OSB+A = OOB + A (in circle BOSG)
= 90°;
.-. SA = SA V
Hence the double point 8 of the two directly similar figures
on AB, AC may be found by drawing the Symmedian chord AAy
through K and bisecting it at iS.
LEMOINE GEOMETRY.
89
121. Let T-^T^Tg, be the Tangent Triangle, formed by drawing
tangents to the ciroumoircle at A, B, G.
Then AT X , jBT 2 , GT t , are concurrent at K.
For, if q, r be the perpendiculars from T t on AO, AB,
q _ TfisYnACTs _ ainB _ b
r ~~ T X B sin ABT S ~ sin G ~ c '
Hence AT X passes through K ; so also do BT. 2 , CT S .
Note that T x is the point of intersection of the tangents at
B and G, whose equations are y/c + a/a = 0, a/a + b/B = 0, so
that the n.c. of 2\ are (—a, b, c).
122. The Lemoine Point A of 1^1,-
In the figure of the preceding section, it will be seen that
the Lemoine Point K of the inscribed triangle ABG, is the
Gergonne Point (32) of T^T,.
Therefore the point M, which is the Gergonne Point of ABG,
is the Lemoine Point of XYZ.
But XYZ and 7^1, are homothetic, the centre of similitude
being <r. (26)
Therefore the Lemoine Point — call it A — of I-JJt lies on a-M,
and a-M: trA = linear ratio of XYZ, Ijljls = r : 2fi.
90 MODERN GEOMETRY.
The point A has n.c. (* — a), ... ; for the perpendiculars from
the point {(s—a), (s—b), (s—c)} on the sides of It.IJL 3 are found
to be proportional to the sides of this triangle.
Note the following list of " (s—a) " points : —
(1) Nagel Point: b.c. are (s—a), (s—b), (s—c). (30)
(2) Gergonne Point : b.c. are lj(s-a), &c. (32)
(3) Lemoine Point A of 1x1^: n.c. are (s — a),
(4) Centre of Sim. o- of XTZ,IJJ S : n.c. are l/(s-a) (26)
123. To prove that AK bisects all chords of the triangle ABG,
which are parallel to the tangent at A, or perpendicular to OA,
or parallel to the side H 2 H S of the orthocentric triangle H^ff,?,.
Let BAK=6, GAK=<j>.
Then sin 0/sin <f> = y//3 = c/b
= sin O/sin B = sin BATjsia GA'L\.
Therefore AK and T 2 AT S are harmonic conjugates with
respect to AB and AG.
It follows that AK bisects all chords which are parallel to
r,T s or H 2 H S , or are perpendicular to OA.
124. The Harmonic Quadrilateral. — The angles of the har-
monic pencil at A are seen to be B, <j>, or 0, 6, <j> (Fig., p. 89).
The same angles are found,in the same order, at B and G and A v
Hence the pencils at B, C, A and A 1 are harmonic; ABA-fl
being called, on this account, a Harmonic Quadrilateral.
In the triangle ABA X the tangent at B is harmonically con-
jugate to BK V so that BK X is the B-symmedian for this triangle.
Similarly, A 1 K l is the j4.,-symmedian in BA X G Y ; and GK X the
C-symmedian in AGA V
To prove that rectangle AB.Afi = rectangle AG.A X B.
sin 0/sin <£ = sin BGA/sm GBA, = A X B/Afi.
.: AB.A.C = AG.A.B.
If x, u, z, u are the perpendiculars from K x on AB, AG, A-Ji,
A X G, then x/AB = y/AG = z/A.B.
For, since AK X is the -4-symmedian of ABG,
.-. x/AB = y/AG.
And, since BK X is the B-symmedian of AABA lt
x/AB = z/AiB.
And, since A^K^ is the J^-symmedian of A X BG,
:. z/A r B = u/AxG.
LEMOINE GEOMETRY.
91
125. The Lemoine Point is the point of intersection of lines
joining the mid-points of the sides of ABO to the mid-points of
corresponding perpendiculars.
Let A', B', O be the mid-points of EG, GA, AB, and let A'K
meet AD in I.
Then, since BK and the tangent BT X at B form a harmonic
pencil with BA, BO, therefore the range (AKK^) is harmonic ;
therefore the pencil A'(AlD<x> ) is harmonic, so that AB is
bisected at I.
A rectangle XYVTJ being inscribed in ABG with the
side XY on BO, to find the locus of its centre P.
The diagram shows that the mid-point of JJV lies on A'A,
and that P lies on A'l.
The Lemoine Point K, common to A'l, B'm, O'n is the com-
mon centre of three inscribed rectangles, standing on BO, GA,
AB respectively.
126. To determine the direction angles 6 lt 2 , S , which OK
makes with the sides of ABG.
Prom the diagram,
OK cos 1 = A'd = A'B-j~.
Now
and
A'D
6 2 -
2a
Kd =
2A
at + V+c*
(117)
ID = ifc. = _
21 a
92 MODERN GEOMETRY.
• cos6 1 = m.a(b' ! -c i ), where 1/m = 0-K\(a 2 + 6 2 + c 2 ) ;
cos^ : cos ft, : cos# s = a(6 2 — c 2 ) : 6 (c 2 — a 2 ) : c(a 2 — 6 2 ).
Hence the tripolar equation to OK is
a 2 (6 2 -c 2 )r 1 2 +6 a (c 2 -a 2 )r 2 2 + c 2 (a 2 -6 2 )r s 2 = 0. (17)
127. The Apollonian Gircles. — Let the several pairs of bisectors
of the angles A, B, G meet BG in a, a' ; GA in /3, /3' ; AB in y, /.
The circles described on aa', /3j3, yy' as diameters are called
the Apollonian Circles.
Let the tangent at A to the circle ABG meet BG in L v
Then angle L x aA = aBA + BAa = B+%A,
LAa = L,AG + GAa = B +%A ;
so that L-fl. = LyA..
And, since aAa! = 90°,
i^ — L x a' ;
so that i, is the centre of the Apollonian Circle (aa'), passing
through A and orthogonal to the circle ABG.
Since Ali x is a tangent, the polar of L x passes through A ;
and, since (JBK-fiL^ is harmonic, the polar of L t passes
through K x .
It follows that AKKAflt is the polar of £, ; so that L t A x is
the other tangent from L v
Hence the common chords AA V BB„ GG X of the circle ABG
and the Apollonian Circles intersect at K, which is therefore
equipotential for the four circles.
Note that OL t bisects AKK 1 A l at right angles, and therefore
passes through the point 8 of Section (120).
128. The Lemoine Axis.
Since the polars of L v L 2 , L t pass through K, therefore L X L 2 L S
lie on the polar of K.
The equation to the tangent at A is /3/b+y/c = 0, &c.
Hence L-Jjjj^ is
a/ a + J3/b + y/c = 0, or z/a 2 + y/V + zj<? = 0.
This is called the Lemoine Axis.
129. A Harmonic Quadrilateral such as ABA-yC can be in-
verted into a square.
LEMOINE GEOMETRY.
93
Let the circle described with centre T, and radius T,J5 or T,C
cut the Apollonian Circle L l in E and E'.
Then, since the tangents OA, OB, OG, OA t to the two circles
are equal, lies on their common chord EE', and
OE.OW = E 2 ;
so that E, E' are inverse points for the circle ABO.
Let AE, BE, GE, A,E meet the circle again in LMNL'.
Then, taking E as pole and VU as radius of inversion (where
n = power of E — E'—OE 1 ), we have
n T „ in n
LM=AB.-
LN = AG.-
'EA.EB' ~" EA.EC/
But BE : EG = BA : AC,
since E is on the Apollonian Circle aAa! ;
.', LM - LN, &c.
Hence LMNL' is a square.
Another square may be obtained by taking E' as pole.
In the above figure, E is on the Apollonian Circle aa' ;
.-. r,:r i = BJE : CE = Ba : aC = c : 6 ;
.-. 6r 2 = cr s ; .'. de = df or /*. = v.
Also A + \ = BEG = Tr-± i .BT 1 C = ^ + A;
.-. X = Jw, so that fx. = v = £-ir.
CHAPTER XI.
LEMOINE-BROCARD GEOMETRY.
130. The Broeard Points. — On the sides BG, GA, AB let triads
of circles be described whose external segments contain the angles
(a) A,B,0;'
(6) G, A, B ;
(e) B, G, A ;
the cyclic order (ABO) being preserved.
The first triad of circles intersect at the orthocentre H.
Let the second triad intersect at O, and the third at O'.
O and li' are the Broeard Points of ABC.
Since the external segment of BOG contains the angle G, this
circle touches GA at G.
Similarly, GilA touches AB at A, and AC1B touches BG at B.
(Memorize the order of angles for Q by the word " GAB.")
In like manner. BCl'C touches AB at B, CQ'A touches BG at
G, AQ'B touches GA at A.
Again, since AQIi touches BG at B, the angle
QBG = QAB.
So QGA = QBG.
Similarly, il'AG - Q!GB = 0,'BA.
Denote each of the equal angles CIBG, QGA, ClAB by m, and
each of the equal angles Q'CB, &c, by <»'.
U
LEM03NE-BR0CARD GEOMETRY. 95
131. To determine u> and <»'.
In the triangle ACIB,
CIB = c sin <o/sin B = 2B sin u>. c/b.
So CIA = 2E sin <o . b/a, QC = 2B sin <o. a/c;
fjhi(A-a>) _ sinfl^O _ ClG _ a 2 .
sinco sin £2 CM CI A be '
.-. coto) = a +6 +c " = cot A + cot B + cotC
4A
_ sin 2 4 + sin 2 B + sin 2 (7 1 + cos A cos B cos C
2 sin 4 sin B sin C sin A sin £ sin f '
The same expressions are found for <o' ;
(■)' = 0).
The angle <o, which is equal to each of the six angles ClAB,
CIBG, CIGA, Cl'BA, Cl'CB, a' AG, is called the Brocard Angle of
ABC.
Since a/ = to, it follows that
Q' A = 2B sin <o . c/a, Q'G = 2 Z? sin a> . b/c, Cl'B = 2B sin ui.a/b.
Observe that the n.c. of K may now be written
2A
a = -3 — 7^ j- a = iatanw = BsinA tanu, &c. (117)
as + V + c- v /
132. To determine the n.c. and b.c. of CI and CI'.
From the diagram,
a = CIB sin cu — 222 sin 2 w . c/b.
So B — 2B sin 2 <o. a/c, y = 2.E sin 2 <o.6/a.
And for CI',
a' = 2R sin 2 (o.6/c, /?' = 2R sin 2 a>. c/a, y' = 2E sin 2 eu. a/6,
.-. oa' = /J/J* = yy' : so that O, 12' are Counter Points.
The b.c. of CI are given by
x:y.z= 1/6 2 : 1/c 2 : 1/a 2 ;
and for CI', x' : y' : a' = 1/c 2 : 1/a 2 : 1/6 2 .
The line CICl' is then found to be
(a 4 -6V).a;/a 2 +(6 4 -c 2 a 2 ). 2 //6 2 +(c 4 -a 2 6- 2 ).z/c 2 = 0.
The power n of O = {x y +y+a)% = w+eV + B y
= i/a 2 + i/V+i/c 2 = n ' ; from s y mmetr y-
.-. oci = oci.
96 MODERN GEOMETRY.
133. The Brocard Angle is never greater than 30°.
For cot o> = cot J. + cot I? + cot C,
and cot B cot G+... = 1 ;
.'. cot 2 <o = cot 2 A + ...+2;
.-. (cotJ3-cotO) 2 +... = 2(cotM+...)-2
= 2(cot 2 co-3).
Hence cot to is never less than a/3, and therefore o> is never
greater than 30°.
134. Some useful formulae.
(a) cosec 2 co = 1 + cot 2 o, = 1 + (a? + V + c 2 ) 2 /16A 2
= (bV + cW + aW)/^ 2 ;
. i , • , a 4 +b 4 + C t -bV-c V-a !! 6'
and 1— 4biit«o = — ! — ; r - 5 —
bV+cW+rfb*
This expression will be denoted by e 2 .
(*)
COS Ci) = j
•2 (feV + cV + trfr) 4
00
. a 2A(a 2 + 6 2 + a
sin 2co = v — — r .
6V+cV+tt 2 fe-
(i)
., . a 4 +6 4 + c 4
COC_a,- 2(b 2 c 2 + ra 2 +aV)'
f*A
nnt .,,.._ » <•* + &«+„«
4 'A(a 2 + 6 2 + c 2 )"
(/) Since sin (J. — co)/sin<o = a 2 /fcc ;
.-. sin (4— co) : sin (B— to) : sin (0— <o) = a 8 : fc 8 : c\
and sin (A— co) sin (7? — to) sin (0— to) = sin 3 to.
(17) sin (4 + w)/sin io = sin A (cot co + cot J. )
= (b 2 + c 2 )/bc.
Note that, when b = c, sin (.A + eo) = 2 am a>.
(h) cos (A + <o)/8in A sinw = cot-4 cotco— 1
= ( - a 2 + 6 2 + c 2 ) (a 2 + 6 2 + c 2 )/16A 2 - 1 ;
.-. cos01+co) = sin A sinco/8A 2 .(6 4 +c 4 — aV-aV)
a sin (A— B) sin B + sin (.4— 0) sin (7.
LEMOINE-BROCARD GEOMETRY.
97
135. Neuberg Circles.
The base BG of a triangle ABG being fixed, to determine the
locus of the vertex A, when the Brocard Angle of the triangle
ABO is constant.
Bisect BG in B : draw BA 1 A i perpendicular, and AM parallel
to BO.
Then
C0t(i) =
AB 2 +AG* = 2. AD 2 + i.BC*
SO 2 + CA 2 + AB°- _ 3a 2 + 4 . AD 2
Take BN = \ a . cot u>
Then
4.areaof ABO " 4a. DM
AZ) 2 -a.D.M".cot<o+3/4.a 2 = 0.
so that L BND = GNB = *>.
NA % = AB^ + NB^-^.BN.BM
= AD 2 -2.£acota>.DM+IW 2
= DN*-3/4..a 2 = 1/4. a 2 (cot 2 o>-3)
= constant.
Hence the locus of 4 is a circle, called a Neuberg Circle,
centre N, and radius p = ^a v cot 2 to— 3-
136. Let BUG, BE'G be equilateral triangles on opposite
sides of the common base BC, so that BE == \a. V3.
Let the Neuberg Circle cut BE in A v A t .
Then DA • #A = DiV 2 - p 2 = 3/4 . a 2 = BE\
And thus, for different values of <a, the Neuberg Circles form
98 MODERN GEOMETRY.
a coaxal family, with E and E' for Limiting Points, and BG for
Radical Axis.
Let GA X cat the circle in T x .
Then, since E is a limiting point,
.-. CA X .GT X = CE 2 = CB\
so that the triangles GBA X , CT,B are similar ;
.-. B'l\ : A X B = BG : A X G.
But A X B = A X G, .: BT X = BG = BE ;
.•. BT X is a tangent at T x .
Similarly, if BA 2 cuts the circle at T 2 ;
then GT„_ is a tangent at T 2 .
137. The Steiner Angles.
From the similar triangles BA X G, T X BG,
z. T x BO = BA X G.
Also, from the cyclic quadrilateral BT X ND X , with right angles
at T x and B, L BT X B — BND or a> ;
.-. z T X DC = B1\B+T X BG = m + BA x C = A x + u;
so that s ™(^ + < ") - sin T* VQ = ^ = 80 = 2
sino) sin BT X B BB BB
a- •! i sin (A, + <u)
Similarly, V-* i = 2.
sin <u
Thus A x , A 2 are the values of x obtained from
sin (x + co) = 2 sin o>.
This gives cot 3 ^a; — 2 cot Ja; . cot to + 3 = 0;
whence cob^A x = cot o> — vcot 2 a> — 3,
cot-J-Aj = cot(o+ veot 2 u>— 3 ;
as is obvious from the diagram.
For . A X D/BB = DN/BB-NAJDB;
cot^A x = cotw — pl\u = cot (0 — vcot 2 o) — 3.
So cot ^j = cot a>+ vcot 2 (u— 3.
The angles \A X , \A^ will be called the Steiner Angles, and
denoted by S x , 8 t .
LEMOINE-BROCAED GEOMETRY.
99
138. Either Brocard Point CI or CI' supplies some interesting
illustrations of the properties of pedal triangles.
Forfl, A+X = BClG = 130 -t' = A + B-
.'. A. = B, so fi = (', v = A.
And for CI', V = (7, / = A, v' = B.
So that the pedal triangles of the Brocard Points are similar
to ABC.
To determine p, the circumradius of the pedal triangle of CI.
ef = 2p sin \ = 2p sin B.
But e/= CIA sin A = 222 sin u.b/a. sin A (131)
.". p = it' sin to.
Hence, the triangles def, ABG have their linear ratio equal to
sinoi : 1.
U = A sin' 2 to.
Also n=4E 2 /A.D";
.-. JS 2 - Ofl 2 = II = 4E 2 sin 2 o),
.-. OK 2 = E 2 (1-4 sin 2 o>) = e ! 2P. (134 a)
139. Lemma I.
Let XBG, YGA, ZAB be isosceles triangles, described all
inwards or all outwards, and having a common base angle 0.
To prove that AX, BY, GZ are concurrent.
Let (ojftyi), (aj&y,)> (a,&y s ) be tne n - c - ° f A - y > ^-
Then 04 = ^ a tan 6 ;
ft = XC sin (0-0) = i.asec0.sin (0-0) ;
y 1 = io sec 6 . sin (B — 6).
A
So that uj : ft
t* 3 : ft
B C
y, = sin : sin {C-6) : sin (2?-0) ;
y 3 = sin (6'— 0) : sin : sin (A— 6),
y 3 = sin (5—0) : sin (A-0) : sin 0.
100 MODERN GEOMETRY.
The equation to AX is /G//3, = //-ft,
or B.ain(B—6) =y sin (G-6),&c.
Hence AX, BY, GZ concur at a point S, the centre of
Perspective for triangles ABC, XYZ, whose n.c. are as
l/sinU-0), l/sin(.B-0), 1/sin (0-0).
The point 8 obviously lies on Kiepert's Hyperbola, the
Counter Point conic of OK, for the K.H. equation is
sin (B— G) . 1/a + . . . — 0. ( Appendix III. a)
140. Lemma II.
The centroid ((?') of XYZ coincides with O.
For if (a, /*, y) be the n.c of G'.
3.a<± = a, + </ 2 + a 3
= ^sec0 {a sind + b sin (0—6) +c.siii(B—0)}
cc 26 sin oc 1/a ; &c.
.•. G' coincides with G. (See Appendix TILb)
141. Illustrations. — (A) In the diagram of Neuberg's Circle
(p. 97), change N into N v and on B0, GA, AB describe the
isosceles triangles N-JBG, N. 2 GA, N a AB (all inwards), with the
common base angle (\ir — to), so that N-^N^N^ are the centres of
the three Neuberg Circles, corresponding to BO, GA, AB.
Then since sin (A — ff) becomes cos (A + (»), the lines AN^BN^,
GN S concur at a point whose n.c. are as sec (A + u>)..., that is,
at the Tarry Point. (143)
For a second illustration take the triangles PBG, QGA,
BAB having the common base angle a> measured inwards.
The triangle FQB, called the First Brocard Triangle, has G
for centroid from (140), while AP, BQ, GB meet at a point D,
the Centre of Perspective for PQB, ABC, its n.c. being as
1/sin (A-o)) Ac, or 1/a 8 , l/b 3 , 1/c 8 , the b.c. being as 1/a 2 , l/b\
1/c 2 .
For the n.c. of /',
a i : P\ '• 1\ =z s i n <° : S1 - n (0 — <") : R i n (B — w ) >
= 1 : c°/ab : b' 2 /ac,
bo that the b.c. of P are as a 2 , c 2 , fe 2 .
LEMOINE-BHOCARD GEOMETRY.
101
Similarly those of Q are c\ b\ a\ and those of E are b\ a\ c a .
The equation of QB is found to be
(a*-bV)m+ (c i -a% i )y+ (b*-cV) z = 0.
This meets BG at a pointy, for which
(c i -aV)y+(b t -cV)z = 0,
or, 2//(6 4 -cV)+V(c 4 -a 2 6 s ) =0.
Hence the Axis of Perspective pqr of the triangles PQR, ABO
is x/(a i -bV) + ... =0. (Appendix III. c)
142. O, 12', K.
Some of the relations between 12, 12', and K, will now be
investigated.
(a) Since JF\4' = BA'.tan PBA' = -|a.tan co = ZJK,
.-. 7fP is parallel to BG,
so TfTQ, 7f 72 are parallel to (74, AB
respectively.
Since OPE = 90° = OQE = OBK,
it follows that P, Q, B lie on the Brocard Circle (OK).
(b) Since the angles PBC, C1BG are each = u>,
.". BP passes through O.
Similarly GQ, AB pass through O, while OP, AQ, BB pass
through J2'.
102 MODERN GEOMETRY.
(c) Since KQ, KB, are parallel to AC, AB,
.-. QPJ2 = QKB = A.
So SQP = B, PBQ = C ;
and thus PQB is inversely similar to ABG, the triangles having
the common centroid G as their double point.
(d) Since
PQB - QAB+QBA = a>+ (B—a>) =B = PQB,
.-. Q (and similarly fi') lies on the Brocard Circle.
(e) lilOK—ilPE=ClBG = <o; soO'O-ET^o).
.-. 0.K bisects Oli' at right angles (at Z).
Prom Section (138).
OO = E(l-4 sin 2 co)» = eE ;
/. OK (diameter of Brocard Circle) = eB sec w.
Oil' = 2 . Oil sin (o = 2eE sin <o.
143. The Steiner and Tarry Points.
The ABC Steiner Point denoted by 2 is the pole of the ABG
Simson Line which is parallel to OK.
To determine 2 geometrically, draw Act parallel to OK,
and o-2 perpendicular to BG. (35)
If V 6 2 , S are the direction angles of OK, it has been proved
that cos 6>, a a(fc 2 -c 2 ). (126)
The n.c. of 2 are 2 B cos 2 cos a , &c, which are as
sec0 1; &c, or as — -^ s ; and the b.c. are as l/(6 2 — c 2 ).
a(o — c)
The point diametrically opposite to 2 on the circle ABG
is called the Tarry Point, and is denoted by T ; therefore the
Simson Lines of 2 and T are at right angles. Hence the n.c.
of Tare 2fl sin0 2 sin0 8 , &c. (46)
Now OK sin 0, = KM- 0A' (KM perp. to BG)
= E sin A tan w—B cos A
oc coa(A+w).
Hence the n.c. of Tare as sec(-4 + w), sec(B+a>), sec(0 + a>).
In (42) let PQB be the Lemoine Axis ; then N is the pole
of the Simson Line parallel to this axis, and therefore per-
pendicular to OK.
Hence N is the Tarry Point.
LBMOINE-BEOCAED GEOMETRY.
103
144. PQB being the First Brocard Triangle, inversely similar
to ABC, to prove that the figure EPBOQ is inversely similar
to 2ACTB.
Since SOT is a diameter, ScrTis a right angle, so that &T is
parallel to BC, and arc Bo- = CT.
Now, since KB is parallel to AB (142a.), and KO to Acr (as
above), .-. z OKB = a-AB = T%C, from the equal arcs.
Similarly, OKQ = T2B ■
also ABG, PQB are inversely similar.
Hence and T are homologous points in these two triangles.
Therefore K and 2 are homologous.
Hence the figures KPBOQ, 2ACTB are inversely similar.
Since L %BA = KQP (similar figures)
= KBP,
and AB is parallel to KB,
.". 25 is parallel to BP, &c.
Hence TA is perpendicular to QB, TB to BP, TC to PQ.
145. Lemma.
The points L, M, N have b.c. proportional to yzx, zxy, xyz,
arranged in cyclic order ; to prove that G', the mean centre of
LMN, coincides with G.
Let (a,/?^). ("Aft). KAys). ( a W) be the absolute n.c. of
L, M, N and G' respectively.
Then, for L,
aajy = bB,/z — cyjx = 2A/(x + y + z).
.-. 04 = 2A/(x + y + z).y/a.
So a. i — 2A/(x + y + z).z/a,
and >*s = 2A/(» + y + z) . x/a ;
3a' = Oj + aj + dj = 2A/a, &c.
Hence G' coincides with G.
104
MODERN GEOMETRY.
146. The b.c. of P, Q, It being aV6 2 , cVa\ &W, in cyclic
order, it follows that G is the centroid of PQB, as already-
proved, and is therefore the double point of the inversely-
similar triangles ABO, PQB.
Let L be the circumcentre of PQB, and therefore of the
Brocard Circle (OK) ; then L in PQB is homologous to in
ABG. Therefore the axes of similitude of the two inversely
similar triangles bisect the angles between GO and GL.
Again, the b.c. of B, O, «' are 1/a 2 , 1/& 2 , 1/c 2 ; 1/& 2 ,
1/c 2 , 1/a 2 ; 1/c 2 , 1/a 2 , 1/6 2 , in cyclic order.
Hence G is the centroid of BCltl'.
Bisect na' in Z, then (2 lies on BZ, and <?£ : GD = 1 : 2.
But 0G:GE= 1:2. (E orthocentre of ABO)
;. BE is parallel to OK, and BE =2.0Z = 2.eB, cos <o.
Let iff meet Dfl" in E'.
Since Zi is the circumcentre, and G the centroid of PQB,
.: LGE' is the Euler Line of PQB.
But LG:GE' = ZG:GB = 1:2;
.". IF is the orthocentre of PQB.
LEMOINE-BROCARD GEOMETRY. 105
Again, HH' = 2 . OL = OK,
.'. H'HKO is a parallelogram
and H'E bisects OH at the Nine-Point centre.
147. Since G is the double point of the inversely similar
figures SAGTB, KPROQ, therefore the points G, 0, T in the
former figure are homologous to G, L, in the latter.
Hence angle OGT = LGO, so that G, L, T are collinear.
Again, if p (=: ^eR sec to) be the radius of the Brocard Circle,
then GO : GL = R: P (by similar figures).
So GT:GO= E: P -
.: GO*=GL.GT;
and GT 2 : GO 2 = R? : p 1 = GT : GL.
148. To prove that D lies on the circumdiameter SOT.
From (146) H'B : ZL - H'G : GL = 2 : 1 ;
and ZL = OZ—p = eR cos o> — \eR sec a> = p cos 2a> ;
.-. H'D -2.ZL = 2 P cos 2o>.
Again, H'T:LT= GT+2.QL-.GT-GL;
and, from above, GT : GL = 5 3 : p' 1 ;
.-. H'T:LT=R? + 2 P l :R i - P !
= 2 cos 2o> : 1 = Jf'D : £0 (or p),
.-. D lies on %0T.
To determine OD.
LT:0T— H'T : LT = 2 cos 2o> : 1,
.-. OD:R = 2cos 2«>-\: 1;
.-. OD = e*R.
Note also that
OD. 02 = e 2 R? = 0« 2 or 0Q"
So that 00, Oii' are tangents to the circles UD1, S1'D%
respectively.
106 MODERN GEOMETRY.
149. The Isodynamic Points.
These are the pair of inverse points 8 and S lt whose pedal
triangles are equilateral; so that
X = fi = v = 60°.
In this case, M = a 2 cot A. + . . . + 4A (64)
= 4A(cotwcot60° + l).
So for S^ M 1 = 4A (cot w cot 60° -1). (68)
The Powers n (n x ) are given by
n (n,) = 8B 2 A/M (Jf,) = 2E7(cota. cot60°±l).
Areas of pedal triangles = 2A 2 /-M" (Mj)
= -|A/(cotft)Cot60°±l).
,, , , / s abc sin (^.±60°)
Absolute n.c = «(«,) = j^ • s \ n6Q c
oc sin (^±60°).
The circumradii of the pedal triangles are given by
2p 2 ( 2pi*) . sin 60° sin 60° sin 60° = area of pedal triangles
= £A/(cotft>cot60°±l).
Let (piPzPs), (p/ftW) bte the tripolar coordinates of 8, 8,.
Then p l sin A = ef = p sin 60° = constant.
•"• Pi : Pa '■ Ps = !/ a : V b '■ V c = Pi' : Pi '■ Pa-
The tripolar equation to OK is
a*(V l -c e )r>+... = 0, for cob0, a a(6 2 -c 2 ) ; (126)
and this is satisfied by r x oc 1/a.
Hence 8, 8, lie on OK.
1 50. Consider the coaxal system which has 8 and 8, for its
limiting points.
From (149), since p : q : r = 1/a : 1/6 : l/o,
therefore the Radical Axis of the system becomes
x/ai + y/tf + z/c 1 = 0,
which is the Lemoine Axis L-^LJL^.
Let this Radical Axis cut OK in X ;
then, since K is the pole of L^L^L^ for the circle ABC,
.'. XO . \K = square of tangent from X to ABO
= AS 2 or X8 X 2 ,
since ABO belongs to the coaxal system.
Therefore the Brocard Circle (OK) belongs to this system,
and therefore is coaxal with ABO.
LBMOINE-BBOCARD GEOMETBY. 107
151. The Isogonic Points.
These are the Counter Points of 8 and S x ; they are therefore
denoted by S' and ?/.
Their antipedal triangles are equilateral, having
areas $M(Af,) = 2A (ootw cot 60° ±1).
Their n.c. are
,, ,, _ abc sin(Bzfc60°)sinC(7±60°)
a *■ ai ; M (M x ) " sin 60° sin 60°
oc l/sin(^±60°).
Hence, from (139), if equilateral triangles XBG, TGA, ZAB
be described inwards on BG, GA, AB, then XA, TB, ZG concur
at 8' ; for the outward system, the point of concurrence is 8/.
These points lie on Kiepert's Hyperbola, whose equation is
sin(J5— 0)/o+... =0.
152. The Circum-ellipse.
Let l/a + m/B + n/y = 0, be the ellipse, with axes 2p, 2q.
Let -aBy, a'B'y' be the n.c. of the centre 12, referred to ABG
and A'B'C, so that 2aa' = — aa + bB+cy.
Project the ellipse into a circle, centre <o, the radius of the
circle being therefoi*e q, while the angle of projection 6 is
oos' 1 q/p.
Let LMN, with angles A/xv, be the triangle into which ABG is
projecbed.
.-. A.Mo>N= A.BQCxcos6;
.'. g 2 sin 2\ = aa X q/p ;
.•. aa = pq sin2A;
so that the ABG b.c. of O are as sin 2A, sin 2ft, sin 2v.
Also ^(aa + bB + cy) = A;
.'. 2pq sin A. sin /t sin v = A.
Again 2aa! = —aa + bB + cy;
aa' = 2pq sin A cos /u cos v ;
so that the A'B'C' b.c. of O are as tan A, tan fi, tan v.
Since A = 2pq sin A sin ju. sin v ;
and — aa+bB+cy = Apq sin A cos/x cos v;
and aa = ^2 sin 2A ;
it follows that
A.( — aa + bB + cy) (aa— bB + cy) (aa + bB—cy)
= 2p i q i . sin 2 2A sin 2 2/u sin 2 2i/ ;
108 MODERN GEOMETRY.
.'. A.^V ( — aa + bB + cy) (aa—bB + cy) (aa + bB-cy)
giving the locus of the centre, when the area of the ellipse
(irpq) is constant.
Let PQB be the triangle formed by tangents to the Ellipse at
A, B, G, then AP, BQ, GB have a common point— call it T —
whose b.c. are as al, bm, en.
Let PQB be projected, into pqr, whose sides touch the
circle LMN.
* Then the projection of T is evidently the Lemoine Point of
LMN (fig., p. 89), and therefore its b.c. are as
MN\ NL\ LM* or as sin 2 A, sin 2 /*, sin 2 * 5
.". al: bm : era = sin 2 A : sin 2 p : sin 2 v.
Hence the Ellipse is
sin 2 A/a./3y+... = 0;
and its Counter Point Locus is
sin 2 A/a.a+... = 0.
This is the Radical Axis of the coaxal system, whose Limit-
ing Points have A/nv for the angles of their pedal triangles.
To calculate the axes of the ellipse in terms of A, fi, v.
From (91),
a 2 cot A + V cot /j. + c 2 cot v = 2A (sec 8 + cos 6)
= 2A (q/p +p/q) = 2A (p» + tf)/pq.
And from (152), A = 2pq sin A sin /a sin v.
... (p + g) * = i /cot*+...+4A ■ x U ■ ■ (84)
sin A. sin/*, sin v sin A sin fi sin v
So (p-q)* = i. ■ x Ml ■ ( 68 >-
sin A sin ju. sin v
. A well known example is the Steiner Ellipse, whose centre is
G, so that LMN is equilateral, and
A = /*'= v — lir.
It will be found that
Z+3* = § ( a 2 +6 2 +c 2 ) = |. A ootco ; pq = 4A/3</3 ;
p/gr = (coteo-f- \/cot 2 <u — 3)/ V3 = cot Sj <\/3 ;
q/p = (cot a) — \/cot 2 a>— 3)/ a/3 = cot S s -v/3 ;
where Sj, S, are the Steiner Angles.
The Counter Point Locus of the Steiner Ellipse is
a/a+B/b + y/c = 0,
which is the Lemoine Axis.
* The point T may be called the Sub- Lemoine Point of the conic.
CHAPTER XII.
PIVOT POINTS. TUCKER CIRCLES.
153. Let BEF be any triangle inscribed in ABG, and let its
angles be A, p, >■.
The circles AEF, BFB, GBE meet in a point— call it 8.
Let def be the pedal triangle of 8, U its area, and^i its
circumradius.
In the circle 8BBF, angle 8DF = 8BF or SBf = Sdf.
So / SBE = Sde ;
d = A., so e = fa., f = v ;
and thus the triangle DS F is similar to the pedal triangle of 8.
The point 8 can be found, as in (56) by drawing inner arcs
(A + A.)... on BG, C A, AB.
Again, LdSf = tt-B = Dtf-f ' (circle 8BBF).
Hence Z d&D = /S-P = eSE.
Denote each of these angles by 6.
Then 8D = a sec 6, 8E - B sec (9, SF = y sec 6,
where (a/fy) are the n.c. of S.
Hence 8 is the double point for any pair of the family of
similar triangles DEF, including def, so that it may be fitly
named the " Pivot Point " (Brehpunkt) of these triangles, which
rotate about it, with their vertices on the sides of ABG, chang-
ing their size but not their shape.
109
110
MODERN GEOMETRY.
The linear dimensions of DEF, def are as sec 0:1; so that,
if M , m are homologous points in these triangles ; then
MSm = 6, MS = mS sec 6 ;
and the locus of M for different triangles DEF is a line through
m perpendicular to Sni.
An important case is that of the centres of the triangles DEF.
These lie on a line through <r , the centre of the circle def,
perpendicular to S<r ; and, if <r is the centre of DBF, then
trScr a = 6.
154. All the elements of DEF may now be determined
absolutely in terms of Xjuv and 0.
For „ = ^. s .HLM+M &c . (65 )
M sin A. '
U=2tf/M;
where M = a 2 cot X+6 2 cot/i + c 2 cot v + 4A ;
and 2p 2 . sin X sin y. sin v = U ;
so that p is known.
Hence SZ* = a sec 0, circumradius of DEF = p = p sec]0,
J5.F = 2p sin X ; area of DEF = U sec 2 = 2tf/M . sec 2 0.
155. The circle DEF cutting the sides of ABG again in D'E'F 1 ,
let A.', ju.', v' be the angles of the family of triangles D'E'F', and
8' their Pivot Point.
In the triangle AF'E,
180°-.! = FX'E+F'FE' = FDF+F'D'E' = X + X',
So 180°-£ = /* + /, 180°-<7 = ■>+><'.
PIVOT POINTS. TUCKER CIRCLES.
Ill
It follows that jS" is the Counter Point of S, and that the
angles of the family D'E'F' are 180°— A— A, .... (102)
The triangles def, d'e'f therefore have the same circumcentre
cr and the same oircnmradius p .
To find o-', the centre of D'E'F', we draw a perpendicular to
iSV S' through o- , and take
Z o-'SV,, = & ;
where ff = d'S'D'.
But <r' coincides with cr, either being the centre of the circle
DD'EE'FF'.
Hence & = 6.
156. To prove that the circle DD'EE'FF' touches the conic
which is inscribed in ABO, and has 8, 8' for foci.
Let 7r be a point where the circle 8<rS' meets the conic.
Then since arc So- = S'cr ;
L S-n-S' is bisected by v<r.
Therefore air is normal to the conic at w.
Now, in the cyclic quadrilateral rrSi?S',
an.SS' - S'cr. Stt +SV.SV = So- (Stt+S'tt) ;
= S(T.2p -. for 2p = major axis of conic.
But SS'/Sa- = 2 . So-JSa- = 2 COS ;
ott = p seoO ■= p.
Also <r is the centre of the circle DD' ... .
Hence this circle touches the conic at t.
112
MODERN GEOMETRY.
157. Triangles circumscribed about ABG.
Through A, B, G draw perpendiculars to S'A, 8'B, S'G,
forming pqr, the Antipedal Triangle of S'.
This triangle (o, S' being fcounter Points) is known to
be homothetic to def, the pedal triangle of S, and therefore to
have angles \, fi, v.
Obviously Sp, Sq, Sr are diameters of the circles BS'Gp, &c.
Through A draw QB parallel to EF, and therefore making
an angle 8 with qr.
Let QG, BB meet at P.
Since AQC = AqG = fi, and ABB = v ;
.-. BPG = X;
so that P lies on the circle BS'Gp.
R_
Hence, as the vertices of DEF slide along the sides of ABG,
the sides of PQB, homothetic to DEF, rotate about A,B, G, and
its vertices slide on fixed circles.
Since S'q is a diameter of S'qQG,
S'Q= S'q cos6, ....
Therefore S' is the double point of the family of triangles
PQB, including pqr.
The linear dimensions of the similar triangles PQB, pqr are
as cos 8:1; so that, if N, n be homologous points in the two
triangles, N describes a circle on S'n as diameter.
PIVOT POINTS. TUCKER CIRCLES.
113
158. To determine the elements of PQB.
From (84) V — area of pqr = %M;
so that, if p' be the circumradius of pqr,
2p n . sin \ sin ft, sin v = \M.
Then, for PQB, circumradius p' = p' cos 6.
Area of PQR = V cos a = £jtf"cos* B.
But area of D E F = 2 A a /M . sec 2 6.
Hence, the area of ABC is a geometric mean between the
areas of any triangle DEF inscribed in ABG, and the area of
the triangle PQB which is homothetic to DEF, and whose sides
pass through A, B, 0.
159. Tucker Circles.
An interesting series of circles present themselves, when
for " Pivot Points " we take the Brocard Points O and li'. The
circle DD'EE'FF', is then called a Tucker Circle, from
B. Tucker, who was the first thoroughly to investigate its
properties.
Since def, d'e'f are now the pedal triangles of O, O' ;
.-. D = d = B, E = e= G, F = f=A;
jy = d'=C, E' = e' = A, F=f = B.
Denote by Z (corresponding to er ) the centre of the common
pedal circle of CI, CI', and by Z' (corresponding to cr) the common
circumcentre of DEF, D'E'F'.
The line of centres ZZ', bisecting CICl' at right angles, falls on
OK; also Z'CIZ =6 = DCld = ECle = FClf.
And since C10Z = a> = ClAF = ClBD = C1GE,
it follows that the figures ClOZ'Z, ClAFf, ClBDd, CiGEe are
similar.
114 MODERN GEOMETRY.
Let OQZ' =■& — AQ.F = BQB = GQE.
Then 6 + ff + a> = \* ;
.-. 6 = \n-v>-&.
Now the radius of the pedal circle of flfl' = p = B sin u>, (138)
.'. circumradius of BB'... = p = p sec 6 = JJ.sin <o/sin (u)-\-ff).
The quadrilateral BDflF being cyclic, BFB = BQB = & ';
therefore the arcs BF', FE', ED' subtend each an angle & at the
circumference, and are therefore equal.
Hence the chord E'B is parallel to AB, F'E to BO, B J F to AG :
and a circle with centre Z' and radius p cos & will touch the
three equal chords.
In the cyclic quadrilateral E'EDF, AE'F=EBF=B; so
that the equal chords are anti-parallel to the corresponding
sides of ABO.
L BE'B' = F'E'B'- F'E'B = A-ff.
F'E'E = F'E'B' + B'E'E -A + ff.
Chord EF' parallel to BO = 2p sin F'E'E = 2,> sin {A + ff) ;
chord BB' cut from B0=2p sin BE'B' = 2,, sin (A-ff).
And if a/3y be the n.c. of the centre Z',
a = p cos %BZ'B' = P cos BE'B' = p cos (A-ff).
PIVOT POINTS. TUCKER CIRCLES. 115
160. The following list of formulae will be found useful.
(a) Radius of circle BB' ... = p = B sin w/sin (a + t)').
(b) Radius of circle touching equal chords = p' = p cos ff.
(c) N.c. of centre Z' ; a = p cos (A— 8').
(d) Length of equal anti-parallel chords = 2p sin &.
(e) Chord BB 1 cut from BG = 2p sin (A-6 1 ).
(/) Chord JE?f" parallel to BG = 2p sin (A + ff).
(g) If d and d 1 are points on OiT such that
OCld = 30°, OQd, = 150°;
then a x cos (A— 30) a sin (4 + 60°),
and <*! oc cos (A - 1 50) «■ sin (A — 60°) .
Hence d and e^ coincide with the Isodynamic Points 8 and h x .
161. The Radical Axis of the Tucker circle (parameter &),
and the circle ABO.
If £/, £ 2 2 , £ 3 2 are the powers of 4, B, .€ for the Tucker circle,
then the required Radical Axis is t 1 i x + ... =0. (62)
Now,
atp An sin 6' o r> • sin 5 sin &
AF = AQ . -; — -^ = 2K Bin (
'sin (a>-|-0') 'sin4'sin((u + 6')
n sin_B . /,,
= 2p . — — - . sin 6'.
smA
And since EF' is parallel to BC,
^ F , = 5^. ^''^JHL^. 2psin(J. + 0');
smi sin .4
. • . ^ ! = 4F. AF' = V- sin 4 sin 5 sin G sin 0'. sin (A + ^)/sin 8 4,
so that the Radical Axis is
8m(A+ff)/a i .x+... = 0.
116
MODERN GEOMETRY.
162. The properties of four Tucker Circles, whose centres
are certain standard points on OK, will now be discussed.
(A) The First Lemoine Circle, or Triplicate Ratio Circle.
This has its centre at L, the mid-point of OK, or the
centre of the Brocard Circle.
so that ff = a).
Then, (a) p = B, sin a>/sin 2a> = ^R sec w.
(b) p' — p cos io = |B.
(o) a = ^R-secw.cos (A— &>).
(d) Anti-parallel chord = M sec to . sin u> = i? tan a>.
(e) Chord Z>D' cut from BG = R sec'w sin (A— u>).
So that
BB : EE' : FF' = sin (A- a) : sin (£- u>) : sin (C-a>)
= a' : ?) 8 : d\
Hence the name " Triplicate Ratio Circle." >
(/) Chord EF' parallel to BO = R sec co.sin (A + u).
These chords pass through K. For since chord EB' is anti-
parallel to AB, EB'G = J,
perp. from B on BG = EB' sin A = II tan w . sin A, from (<£)
s= perp. from JC on #0. (131)
PIVOT POINTS. TUCKER CIRCLES 117
163. (B) The Pedal Circle of OO'.
The centre being Z, 6' = %ir— co.
(a) p =: B sin to.
(6) p'=B sin 2 a..
(c) a — B sin co. sin (^L-t-co) = B sin 2 co. J b/c+e/b].
(d) Anti-parallel chord = B sin 2u.
(e) Chord cut from BG = 2B sin co cos (X+co).
(/) Chord parallel to BG = 2B sin co cos (.4— to).
164. (C) The Second Lemoine Circle, or Cosine Circle.
A
The centre of this circle is K, so that & = OCIK = \tc.
(a) p = B tan co.
(6) P ' = 0.
(c) u. = B tan co sin .4. (131)
(d) Anti-parallel chords E'F, F'B, B'JE each equal
the diameter 2B tan co ; so that they each
pass through the centre K, as is also obvious
from (6).
(e) BB' = 2R tan co cos A ;
.-. BB' : BE' : FF' = cos A : cos B : cos G.
Hence the name " Cosine Circle."
(/) Chord parallel to BG = 2B tan co cos A
= chord cut from BG.
118
MODERN GEOMETRY.
165. The Taylor Circle.
Let H u .Hj, Sj be the feet of the perpendiculars from
A, B, on the opposite sides.
Draw H^F perpendicular to AB, H^D to BO, H S E to CA.
Let
Then
and
AQF = 4>.
AF = AH l sin B = 2B sin 2 B sin G
AQ = 2B sin o sin -B/sin A ;
sin (u) + cj>) _ J.O _ sin <u
_
Also
Hence
sin<£
COt 0) =
sin A sin B sin '
1 + cos A cos B cos (7
(131)
(131)
sin A sin £ sin G
tan <£ = —tan A tan 2? tan G.
Similarly it may be shown that
tan BQD or tan GQF = — tan A tan B tan ;
so that AQF = BQD — OQE.
Next, draw H^E' perpendicular to GA, H 2 F' perpendicular to
AH, Kjy perpendicular to BC.
Then it may be shown that
AQ'E' = BQ'F' = GQ'B' = <£.
The six triangles AQF, ... being all similar, it follows that
BB'EB'FF' lie on a Tucker circle, called the Taylor circle, after
Mr. H. M. Taylor.
A
B D'~ " D
The angle <j> is called the Taylor Angle.
PIVOT POINTS. TUCKER CIRCLES. H9
Since <f> is less than it, we have
D sin ^> = + sin A sin B sin G, D cos <f> = —cos ^1 cos B cos C,
where _D 2 = cos 2 A cos 2 S cos 2 G + sin 2 4 sin 2 B sin 2 0.
A diagram shows that, T being the centre of this circle,
OT : TK — —tan <p : tan a> = tan A tan B tan G : tan w.
Note the equal anti-parallel chords DF', FE', ED' ; also the
chord E'D, parallel to AB, F'E to BG, D'F to AG.
166. The list of formulas is now —
(a) Radius of T-circle = R . s ™ w N = RD.
sin (a> + <£)
(6) p' = RD cos <£ = i2 cos JL cos 5 cos G.
(c) .* = B.D cos (^— <£)
= R (cos 2 .4 cos B cos G— sin 2 JL sin B sin C).
(d) Anti-parallel chord E'F or F'D or D'S
= 2RD sin <£ = 2 B sin 4 sin B sin (7.
(e) Chord cut from BG = 2RD sin (.4— 0)
= R sin 2 A cos (B—G).
(/) Chord .F'E parallel to BG = R sin 2A cos 4 ;
the other chords being D'F and F'D.
To determine the Radical Axis of the circle ABC and the
Taylor Circle.
sin (A + cf) — s i n -^ • cos "£ + cos -4 sin <£
= D(— sin A. cos J. cos 2? cos G
+ cos 4. sins 4 sinB sin G)
cc sin A cos 2 A
Hence, from (61), the Radical Axis is
cotM.a;+... = 0.
So that the tripolar coordinates of the Limiting Points of
these two circles are as cot A : cot B : cot G. (21)
APPENDIX I.
Let LMN..., L'MN ... be two systems of ra points.
Place equal masses p, p at L, L' ; q, q at M, M', &c.
To determine the condition that the two systems shall have
the same mass-centre.
Project LMN... and the mass-centre on any axis, and let
lmn...sc be the distances of these projections from a given point
on the axis.
Then (p + q + r + ...)x = pl + qm,-{-rn + ...
(p + q + r + ...)x — pl' + qm' + rnf + ....
.-. p(l — l')+q(m—m')+r(n—ri) + ... =0.
and so for any number of axes.
But I— I', &c, are the projections of LL', MM.', NN'....
Therefore the required condition is that a closed polygon may be
formed, whose sides are parallel and proportional to p.LL\ &c.
In the case of a triangle
p. LL' a sin (MM', NN').
Now, in the case under discussion, take a second point P on
TT', and let its pedal triangle be d'e'f.
Here L V - dd' = PP . cos 0„
and angle (MM', NN') is (ee', //') or A ;
.'. p cos 0, a sin^., or p a sin A seoO v
Hence all pedal triangles def of points P on TT' have the
same mass-centre for the constant masses sin A sec 6 V See.,
placed at the angular points d, e, f.
120
APPENDIX II.
To determine the second points in which the four circles
cut the Nine-point Circle.
Let the circle PQR cut the sides of ABC again in P„ 'Q„ -B,.
Then AQ.AQ, = AR.AR„ &c;
.-. AQ.AQ 1 .BR.BR 1 .CP.CP l = AR.AR 1 .BP.BP 1 .CQ.CQ,.
But, by Ceva's Theorem, since AP. BQ, OR are concurrent,
BP.GQ.AR = GP.AQ. BR ;
.-. OP, .AQ,.BR 1 = BP 1 . CQ, . AR,.
Therefore AP V BQ U CR r are concurrent.
Again, since CQ : QA = r : p,
and AR : RB = p : q,
.-. AQ=p/(r+p).b; AR=p/(p+q).c.
So AQ 1 =pJ(r l +p i ).b; AR 1 =p i /(p l + q l ).c.
But AQ.AQ 1 = AR.AR 1 , Ac;
. PPi-V _ m ■ c*
(»■ +p) (n +jpi) O ->• q) (p, + ?,) '
(q + ^)(qi + r i) (r+p)(r l +p 1 ) (p + q)(p l + q l )'
121
122 MODERN GEOMETRY.
.*. j>! a — -\ ■ 1- ■
q + r r+p p + q
So 2j oc H h ; r, ex + 4 — ;
.-. g.+r, a 2.<t/(q + r) : q x -r y a 26 , /(r+ 1 »)-2c , /(j> + 2).
In (80) it was shown that the circle PQB cuts the Nme-
Point Circle at a point m, whose b.c. are as a 2 /(g a — i 3 ),
Similarly the circle P 1 Q 1 R 1 (the same circle) cuts the Ninef-
Point Circle at a point <»', where the b.c. of m' are given by
, &c.
(Si + nXSi-r,) V(p + q)-c i (p + r)
So, if the circle PQ'R' cuts the Nine-Point Circle again at a lt
the b.c. of o), are given by
^(-p+^-c^-p+r)'
writing — p for -fp.
APPENDIX III.
(a.) To determine the area of XYZ.
Since
A'O = R cos A, A'X — \a tan 6 — B sin A tan 6.
.: 0X = R/cos6.cos(A+6).
.: 2. area YOZ - OY.OZ.smA.
= R- /cos 2 6. cos (B + &) cos (C+6) sin A.
.: 2.AXYZ = -2(Y0Z + Z0X+X0Y) = ... ;
and by some easy reduction we obtain,
AXYZ = A/4 cos 2 6. {2 sin a.— sin (20+ai) }/sin w.
When 6 is equal to either Steiner Angle, (137)
then sin(20 + u) = 2sina>,
and the triangle XYZ vanishes, so that XYZ is a straight
line.
But this triangle always has G for its centroid.
Hence, in this particular case, XYZ passes through G.
(b) Instead of the base angles being equal, suppose that
L XBC = YGA = ZAB = 0,
BOX = CAY = ABZ = 0,
and BXC = CYA = AZB = X -
Then u t = a. sin sin <£/sin x,
Oj = 6 sin<£.sin(0— 0)/sinx,
Oj = c sin 0. sin (S — <£)/sin x ;
3a = a 1 + «2 + a 8 == ''u & c -
Hence G is the centroid of XYZ.
123
124 MODERN GEOMETRY.
(c) Let YZ, ZX, XY meet BG, OA, AB in x, y, z respect-
ively.
Then, since AX, BY, GZ are concurrent, xyz is a straight line,
being the axis of perspective of the triangles ABC, XYZ.
To show that the envelope of xyz is Kiepert's Parabola.
The equation of yz is
(Ay»— Ays) a + (y 2 a s— y^) P+ (<hfi»— a sA) y = 0.
Now y 2 a 3 — y 3 a 3 a sin (A — 8) sin (-B— 6) — sin 6 sin ;.
x sin J. sin B— sin sin 2ft
So <hP&~ «sA a s i n C sin ^1— sin I? sin 26.
Therefore at x we have
/?/(sin sin^l — sinS sin 26) +y/(&mA sin B— sin (7 sin26) = 0,
so that xyz is
aa/(sin.Asin.Bsin'C ! — sin 2 4 sin 26) + . . . = 0.
Writing this as px + qy+rz — 0, we know, from (9), that
this line touches the parabola, the n.c. of whose focus are
o/(l/ S -l/r), &c.
Here 1/q— 1/r a (sin 2 B-sin 2 0) sin 26.
Hence the focus has n.c. a/(b 2 — c 2 ) &c, and the directrix is
(fe 2 -c ! )cos4-a+... =0.
Hence the envelope of xyz is Kiepert's Parabola, having for
focus the point whose Simson Line is parallel to OGS, and
OGH for directrix.
INDEX.
[The numbers refer to Sections.]
Aiyar, V. R., 108, 109.
Angular Coords., 81.
Apollonian Circles, 127, 129.
Artzt's Parabola, 66, 120.
Axis of Perspective :
ABC and PQB, 141.
,, X¥Z, App. III(c).
Beard, W. P., 36.
Brocard Circle, 142, 144, 150.
Brocard Angle and Equil. T., 91.
Centres of Similitude :
XYZ and I,/^, 26.
defa.niD'E'F 1 , 85.
H&H* and T^T*, 85.
Circles :
Apollonian, 127, 129.
Brocard, 142, 144, 150.
Centre, 9, 12, 42.
Lemoine (First), 162.
(Second), 164.
Neuberg, 135.
Nine Point, 16, 45, 61, 107.
Taylor, 165.
Pedal C. of nn', 163.
Centre lt 44.
Cubic Transformation of Elliptic
Functions, 34, 54, 55.
Davis, R. F., 9, 19, 104.
Dixon.A. C, 55.
1
Feuerbach Point, 14, 51.
,, Theorem, 63.
Gegenpunkte, 83.
Genese, Prof. R. W., 102.
Gergonue Point (ABC), 32, 34.
„ (T&&), 122.
Greenhill, Sir George, 22, 26, 34,
54, 55.
Harmonic system of lines, 80.
Harmonic Quad., 124, 129.
Isodynamic Points, 99, 149, 160 (g).
Isogonal Conjugates, 101.
Isogonic Points, 151.
Kiepert's Parabola, 11, 40, App.
Ill (c).
Kiepert's Hyperbola, 139, 151.
Lemoine Point of Iiljls, 122.
Lemoyne's Theorem, 77.
Lhuillier, 92.
Limiting Points, 21, 57.
Lines, harmonic system of, 80.
M'Cay's Cubic, 109.
Nagel Point, 30, 31.
Narayanan, S., 75.
• Neuberg, Prof. J., 69, 70, 88, 92,
; 135.
', Nine Point Circle, 16, 45, 61, 107.
25
126
MODERN GEOMETRY.
Orthologio Triangles, 82.
Orthopole is * for TOT, 74.
Parabola : Artzt, 66, 120.
Kiepert, 11, 40.
Points :
SxS-pHiGi (poristioally fixed),
24-28.
FGHO'I'NM {on poristic circles),
29-34.
SEfSiSi' (relations), 87.
Feuerbach, 14, 51. .
Gergonne, 32, 34, 122.
Tsodynamic, 99, 149, 160 (g).
Isogonic, 151.
Lemoine Point of Iil^I-i, 122.
Limiting, 21, 57.
Midpoint of S'S,', 115.
Nagel, 30, 31.
Pivot, 153.
Steiner, 143, 144.
Sub-Lemoine, 152.
Tarry, 141, 143, 144.
Twin Points, 87.
Point D, 141, 146, 148.
Pivot Points, 153.
Poristic formulae, 33.
-Quadrilateral, Harmonic, 124, 129.
Radical Axis :
ABC and Tucker Circle, 161.
,, „ Taylor „ 166.
Groups of E.A.'s, 21, 62.
Rao, T. Bhimasena, 72.
Rouse, E. P., 104.
Schick, Dr. J., 58, 64.
Schoute, Dr. P. H., 100.
Steiner : Angles, 137.
Ellipse, 152.
Point, 143, 144.
Tarry Point, 141, 143, 144.
Taylor, H. M., 103, 106.
„ Circle, 165.
Tricusp.«hyp., 52.
Twin Points, 87.
Weill Point G„ 28.
London : Printed by C. P. Hodgson & Son, 2 Newton Street, Kingsway, W.C.
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