58 
ON A LOCUS IN RELATION TO THE TRIANGLE. 
[394 
We know that if the sides of two triangles touch one and the same conic, their 
angles must lie in and on the same conic. The coordinates of the angles are (1, 0, 0), 
(0, 1, 0), (0, 0, 1) and {A, H, G), (H, B, F), (G, F, C) respectively, and the angles will 
he situate in a conic if only 
1 
1 
1 
A ’ 
H' 
G 
1 
1 
1 
H’ 
B’ 
F 
1 
1 
1 
G ’ 
F ’ 
G 
an equation which must be equivalent to the last preceding one; this is easily verified. 
In fact, writing for shortness 
1 
1 
i 
, □ = 
1 
1 
1 
a ’ 
h ’ 
9 
A ’ 
H’ 
G 
1 
1 
1 
1 
1 
1 
h * 
b ’ 
/ 
H ’ 
1 
1 
1 
1 
1 
1 
9' 
7’ 
c 
’ 
F’ 
O 
we have 
“ n = ABCF• (B ° + CFGH- (FG ~ 1:11 * + BfV-II 1 ,!G> ' 
= jBcFwm (aG ‘ №+hABFG +9 CAH n 
and the second factor is 
= ctGH (AF + Kf) + AFhBG + AFgCH, 
= AF(aGH -t- hBG + gCH) + KafGH. 
But 
aGH + hBG + gCH = G (aH + hB) + gCH = G - gF + gCH, 
= G-gF+gCH, 
= -g(FG-GH), 
= -ghK, 
so that the second factor is 
= K (a/GH - ghAF), 
which is 
= K (f -g-h 1 — bcg' 2 h 2 — cab? f 2 — abf -g 2 + 2abc/gh), 
= Kabcf-g-h 2 V,