54 
ON A LOCUS IN RELATION TO THE TRIANGLE. 
[394 
(A, H, G), (H, B, F), (G, F, C). Hence considering a point P, the coordinates of which 
are (x, y, z), and taking (X, Y, Z) for current coordinates, the equation of the perpen 
dicular from P on the side X = 0 is 
X, Y, Z 
X , y , z 
A, H, G 
= 0, 
and writing in this equation X = 0, we find 
( 0 , Ay — Hx, Az — Gx) 
for the coordinates of the foot of the perpendicular. For the other perpendiculars 
respectively, the coordinates are 
(Bx — Hy, 0 , Bz — Fy), 
and 
(Gx -Gz, Cy-Fz , 0 ), 
and hence the condition in order that the three feet may lie in a line is 
0 , Ay — Hx, Az — Gx = 0; 
Bx — Hy, 0 , Bz — Fy 
Gx — Gz , Gy — Fz, 0 
or, what is the same thing, 
(Ay — Hx) (Bz — Fy) (Gx — Gz) + (Az — Gx) (Bx — Hy) (Gy — Fz) = 0, 
that is 
2 (ABG - FGH) xyz 
+ A( FH- BG)yz 2 + A (FG — CH) y-z 
+ B( FG - CH) zx* + B (GH — AF) z-x 
+ C( GH- AF)xy* + C (HF-BG)a?y = 0, 
which is the equation of the locus of P; the locus is therefore a cubic. Writing 
for a moment 
(BG-F\ GA - G-, AB-1F, GH-AF, HF-BG, FG - CH) = (A', B', G', F', G', H), 
and K' for the discriminant ABG — AF-— &c., the equation is 
2 (ABG — FGH) xyz + Ayz (H'y + G'z) + Bzx (Hx + Fz) + Gxy (G'x + Fy) = 0, 
or as this may also be written 
FG'H (ABC ~ FGH) xyz + F yz {n + h) + G’ ZX (p + h) + H xy (p + I') = °’