364] ON A THEOREM RELATING TO FIVE POINTS IN A PLANE. 481 
that is 
x( YZ' — Y'Z) + y { Z'X) + z(— XY' ) = 0, 
x (— YZ' ) + y ( ZX'-ZX) + z( X'Y) = 0, 
x( + Y'Z) + y(-ZX' ) + z( XY' — X'Y) = 0, 
which are obviously the equations of three lines which meet in a point. 
But the theorem may be exhibited as a theorem relating to a quadrangle 1234 
and a point O'; for writing 1, 2, 3, 4 in place of A, B, G, 0, the triangle A'B'G' is 
in fact the triangle formed by the three centres 41.23, 42.31, 43.12 of the quadrangle 
1234, hence the triangle in question must be similarly related to each of the four 
triangles 423, 431, 412, 123; or, forming the diagram 
P 
Q 
R 
8 
41.23 
4 
3 
2 
1 
42.31 
3 
4 
1 
2 
43.12 
2 
1 
4 
3, 
of the theorem : 
viz. 
the 
lines 
«4, 
/33, 7 2 meet 
in 
a point 
p 
a3, 
/34, 7 1 
„ 
yy 
Q, 
«2, 
/31, 7 4 
yy 
yy 
p, 
al, 
/32, 7 3 
yy 
„ 
8, 
or, what is the same thing, we have with the points 1, 2, 3, 4 and the point 0' 
constructed the four points P, Q, R, S such that 
1$, 2R, 3Q, 4<P meet in a point a, 
2S, IP, 4Q, 3/3 „ „ /3, 
3$, 4P, IQ, 2P „ „ 7 . 
The eight points 1, 2, 3, 4, P, Q, P, S form a figure such as the perspective 
representation of a parallelopiped, or, if we please, a cube ; and not only so, but the 
plane figure is really a certain perspective representation of the cube; this identi 
fication depends on the following two theorems: 
C. Y. 
61