57 
394] ON A LOCUS IN RELATION TO THE TRIANGLE. 
being satisfied, the cubic breaks 
up into the line 
x y z 
+ - + t = 0, and the conic 
f 9 h 
A B G 
~7 yz + — zx + y- xy = 0. 
I 9 h 
It is to be remarked that in general a triangle and the reciprocal triangle are 
in perspective; that is, the lines joining corresponding angles meet in a point, and 
the points of intersections of opposite sides lie in a line; this is the case therefore 
with the triangle (x = 0, y = 0, z = 0), and the reciprocal triangle 
(ax+ hy+gz = 0, lix + by +fz = 0, gx +fy + cz = 0); 
and it is easy to see that the line through the points of intersection of corresponding 
sides is in fact the above mentioned line % + - + ^ = 0. It is to be noticed also that 
J 9 h 
the coordinates of the point of intersection of the lines joining the corresponding 
angles are (F, G, H). The conic 
A B G 
-tyz-\— zx + y- xy = 0 
/ 9 h 
is of course a conic passing through the angles of the triangle (x = 0, y = 0, z — 0); 
it is not, what it might have been expected to be, a conic having double contact with 
the Absolute (a, b, c, f, g, li\x, y, z)\ 
I return to the condition 
1 _ 1 _ 1 _ 1 2 
abc of 2 bg 2 cJi 2+ fgh ’ 
this can be shown to be the condition in order that the sides of the triangle 
(x = 0, y = 0, z = 0), and the sides of the reciprocal triangle {ax + liy + gz = 0, hx+ by +fz = 0, 
gx + fy + cz = 0) touch one and the same conic; in fact, using line coordinates, the 
coordinates of the first three sides are (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively, and 
those of the second three sides are (a, h, g), (h, b, f), (g, f, c) respectively; the equation 
of a conic touching the first three lines is 
L M N 
z + - + -p = 0, 
ç V s 
and hence making the conic touch the second three sides, we have three linear 
equations from which eliminating L, M, N, we find 
1 
1 
1 
a ’ 
1’ 
9 
1 
1 
1 
h’ 
6’ 
7 
1 
1 
i 
9' 
/’ 
c 
which is the equation in question. 
C. VI. 
8