424 
[857 
857. 
ANALYTICAL GEOMETRICAL NOTE ON THE CONIC. 
[From the Messenger of Mathematics, vol. xv. (1886), p. 192.] 
Take (X, F, Z) the coordinates of a point on the conic yz + zx + xy = 0, so that 
YZ + ZX + XY = 0; clearly (F, Z, X) and (Z, X, Y) are the coordinates of two other 
points on the same conic; I say that the three points are the vertices of a triangle 
circumscribed about the conic 
x 2 -\-y 2 + z 2 — 2yz — 2 zx — 2 xy = 0. 
In fact, the equation of one of the sides is 
X , y , Z 
= 0, 
X, Y, Z 
Y, Z, X 
say this is AX + BY+CZ=0, where A, B, C = XY-Z\ YZ - X 2 , XZ-Y 2 ; and the 
condition in order that this side may touch the conic 
is 
But we have 
x 2 + y 2 + z 2 — %yz — 2 zx — 2 xy = 0 
BG +CA + AB = 0. 
BG + CA + AB = Y 2 Z 2 + Z 2 X 2 + X 2 Y 2 - X (Y 3 + Z 3 ) - Y(Z s + Z 3 ) - Z(X 3 + F 3 ) 
+ X 2 YZ + XY 2 Z + XYZ* 
= (YZ+ ZX + XY) (— X 2 — Y 2 - Z 2 + YZ + ZX + XY) = 0 ; 
and similarly for the other two sides. The point (Z, F, Z) is an arbitrary point on 
the conic yz + zx -{■ xy — 0; and we thus see that we have a singly infinite series of 
triangles each inscribed in this conic and circumscribed about the conic 
x 2 + y 2 + z 2 — 2 yz — 2 zx — 2 xy = 0.