480 
[364 
364. 
ON A THEOREM RELATING TO FIVE POINTS IN A PLANE. 
[From the Philosophical Magazine, vol. xxix. (1865), pp. 460—464.] 
Two triangles, ABC, A'B'C' which are such that the lines AA', BB', CC meet 
in a point, are said to be in perspective ; and a triangle A'B'C, the angles A', B', C' 
of which lie in the sides BG, G A, AB respectively, is said to be inscribed in the 
triangle ABG ; hence, if A', B', G' are the intersections of the sides by the lines 
AO, BO, GO respectively (where 0 is any point whatever), the triangle A'B'C' is said 
to be perspectively inscribed in the triangle ABG, viz. it is so inscribed by means of 
the point 0. 
We have the following theorem, relating to any triangle ABG, and two points 
0, O'. If in the triangle ABG, by means of the point 0, we inscribe a triangle A'B'C7, 
and in the triangle A'B'C', by means of the point O', we inscribe a triangle a Ay, 
then the triangles ABG, afiy are in perspective, viz. the lines Aa, B/3, Gy will meet 
in a point. 
This is very easily proved analytically ; in fact, taking x = 0, y = 0, z — 0 for the 
equations of the lines B'G', C'A', A'B' respectively, and (X, Y, Z) for the coordinates of 
the point 0, then the coordinates of (A, B, G) are found to be {—X, Y, Z), (X, — Y, Z), 
(X, Y, — Z) respectively. Moreover, if {X, Y, Z') are the coordinates of the point O', 
then the coordinates of (a, A, 7) are found to be 
(0, F, Z'), (X, 0, F), {X, Y, 0) 
respectively. Hence the equations of the lines Aa, BA, Gy are respectively 
X , 
y > 
2 
= 0, 
x , 
V’ 
z 
= 0, 
x , 
y > 
z 
-x, 
F, 
Z 
X, 
-Y, 
Z 
X, 
Y, 
-Z 
0, 
F, 
Z' 
X', 
0, 
Z' 
X', 
Y, 
0 
= 0;