82 The Right Line.

POINTS.

68. If a, (3, 7 le the normal co-ordinates of a point M, the
f^int JT. ichose co-ordinates are j3 -f 7, 7 + a, a -f /? is called
the siisvlixu-ntary of M.

*' F y

/? 4- 7 7 -f a a-!-/?

Fence, if we seek whether M, M' can coincide, we must have
a j3 7 a + /? + 7

jS-ry y-ra a+0 2 (a +/2-h 7)
These will be satisfied either by a = {$ = 7; that is, by the
ineentre of the triangle of reference, or by the points of the
Hue a -f /3 ~ y = 0, which is the trilinear polar of the ineentre.
EXEBCISES.
1. Any point and its supplementary are collinear with the ineentre.
2. If Jf deserlbe the line la ~ mB -r ny = 0, prove that If describes
(/ 4- in -f «) (a -f ^ -f 7) - 2 (la + »2j8 + My) = 0, (197)
S. The points supplementary to the summits of the triangle of reference
are tie points A', £', C', where the internal bisectors meet the opposite sides.
For, putting » = 0 in (197), "vre see that the supplementary of any line
la -r «j8 = 0 passing through C is the line (I — m)(a — j8) - (I + m) y passing
through €'.
4. The supplementary of the triangle whose summits are the centres of
the escribed circles is the triangle of reference.
TEIA^GLES ET MULTIPLE PEESPECTITE.
69. "We hare given, in § 59, the fundamental property of
triangles in perspective; but here we shall enter into more
detail.
To find the condition that the triangle of reference may "be in
perspective with one whose summits have the co-ordinates
a2/32y2, 03)8373, or whose sides have the equations
Jia -f «!/J 4 «i7 = 0, Z2a -f w2/5 + mzy = 0, lza + m9fi + ny3 = 0.
1°. The equations of the joins of corresponding summits are
easily found to be /?/& « y/7l; 7/70 = a/az; a/a, = /?/&. Hence,
eliminating, the condition of concurrence is
(198)