394] ON A LOCUS IN RELATION TO THE TRIANGLE. 63
that is, at each of the points B, A, C\ The right-hand side of the foregoing
equation is
— — (ab — h )(ha, ab, lib^x, y) 2 , = — (ab — li*) h ^ax 2 + by 2 + xy^j ,
so that the equation may also be written
(ax + hyf (fix + by) 2 + ch*z* (ax* + by* + 2 |- xy\ = 0.
Secondly, to eliminate the (x, y, z), we have
_ab — h* Y 2 _ 1 aa + lift 1 ha + b/3
hub Z h a ha+ b/3 b aa + h/3~ ’
where
Y __ c/iy 2
I = _ (aa + hj3)(ha + b0) ’
or, completing the elimination,
(ab — h*) chy 2 = (/¿6, — ab, ha\aa + lift, ha + b/3)*
= — (ab — h*) h (aa* + bj3 2 + afi'j ,
that is
(«, b, c, 0, 0, ~Ja, /9, 7^ = 0.
Writing (a;, y, in place of (a, ¡3, 7), the locus of the point 0 is the conic
(a, b, c, 0, 0, yjja, y, zj* = 0,
which is a conic intersecting the Absolute
(a, b, c, 0, 0, li§x, y, zf = 0,
at its intersections with the lines x = 0, y — 0, that is the lines C'B' and O'A'.
In regard to this new conic, the coordinates of the pole of C'B' (x = 0) are at
once found to be (— li, a, 0), that is, the pole of C'B' is B; and similarly the coordi
nates of the pole of C'A'(y = 0) are (b, — h, 0), that is, the pole of C'A' is A. We
may consequently construct the conic the locus of 0, viz. given the Absolute and the
points A and B, we have C'A' the polar of B, meeting the Absolute in two points
(a 1} a 2 ), and C'B' the polar of A meeting the Absolute in the points (b, and b 2 ); the
lines C'A' and C'B’ meet in C\ This being so, the required conic passes through the
points tq, a,, b u b 2 , the tangents at these points being Aa x , Aa 2 , Bb x , Bb 2 respectively;
eight conditions, five of which would be sufficient to determine the conic. It is to be
remarked that the lines C'B', C'A' (which in regard to the Absolute are the polars of
A, B respectively) are in regard to the required conic the polars of B, A respectively.
The conic the locus of 0 being known, the point O may be taken at any point
of this conic, and then we have A' as the intersection of C A with AO, B as the
intersection of C'B' with BO, and finally, C as the pole of the line A B in regard
