Systems of Three Co-ordinates. 81

3. Find the area of the triangle formed hy the lines

IICL -f Wi/3 + n\y = 0, ha + mz& + nyy = 0, ?3« + 013$ -f n%y = 0.
Solving hetween the second and third, we get the co-ordinates of their
point of intersection proportional to the minors Zi, MI, Ni of the determi-
nant (Jiwz2»3). Hence, in this case,

Ti = Xi sin -4 + Mi sin 2) + JVi sin C, &c. ;
and substituting in equation (191), we get the area.

4. If (AI, jj.1) V}); (AS, fizj vz) ; (As* /*39 ^s) he the ahsolute harycentric
co-ordinates of three points, prove that the area of the triangle whose sum-
mits they are is A (

COMPLEMENTARY POINTS AND FIGURES.

67. Let A', £', Cr le the middle points of the sides £0, CA,
AB of the triangle of reference. Then, if M, M1 le homologous
points with respect to ABC, A'B'G', M' is called the complemen-
tary of Mj and M the anti-complementary of M'.

If G be the centroid of ABC, then it is also the centroid of
A'B'C' '; that is, it is their double point. Hence G divides
MM' in the ratio 2:1. Hence if (a/5y), (a'/3y) be the
absolute barycentric co-ordinates of M, M', the co-ordinates of
# are~" '

Hence a = ? ^.f y-, (194)
a = P' + y-af, /3 = a'-/3' + y', y = af + $' -yf. (195)
If the point M describe any figure F, Mr will describe a
figure Ff . F' is called the complementary of F, and F the
ant i-complementary of -F'.
EXERCISES.
1. If three concurrent lines he drawn through the middle points of the
sides of a triangle, parallels to them through the summits are concurrent.
2. If A\S\0\ he the triangle formed by parallels to BC, GA, AS through
Ay J5, Cj the triangles AiB\C\, AJBO have M} M' as homologous points.
3. In normal co-ordinates, the complementary of the point afty is the point
b$ + cy cy + aa aa + b$ 1>& -f cy - aa,
— o — » — OA — 9 — o - ' ^e anti-complementary, the point - - - — ,
2i(i Lt) 2c &
&c. (196)
4. Centre of circle AM 0 is complementary of orthocentre.