The One Circle - Two Point Problem of Apollonius
- Dear Hyacintthists,
Let ABC be a triangle, P=(u:v:w) a point and DEF the cevian triangle of P.
The radical axes of the circumcircle and arbitrary circle passing through E and F intersect at a point A' on the line EF.
If A' is outside the circumcircle then there are two circles which pass through E and F and are tangent to the circumcircle. The contact points are the intersection of circumcircle with the polar p_a of A' with respect to the circumcircle.
Similary construct B', C' and p_b, p_c.
A'=(u(b^2w(u+v)-c^2v(u+w)): b^2v*w(u+v): -c^2v*w(u+w))
The points A', B' and C' are aligned in the trilinear polar of
T = u / (a^2v*w(u+v)(u+w) - u(v+w)(b^2w(u+v)+c^2v(u+w)) : ... : ...
Therefore, the lines p_a, p_b and p_c are concurrent in the pole of A'B' with respect to the circumcircle; its barycentric coordinates are:
Q = a^2(a^4v^2w^2(u+v)(u+w) +
a^2u*v*w(u^2(b^2w+c^2v) - v*w(b^2v+c^2w)) -
u^2(v+w)(c^2v-b^2w) (u(c^2v-b^2w)-v*w(b^2-c^2))) : ... : ...
A few examples of triples of points (P, Q, T)
(X(1), -- , X(267)); (X(2), X(25), X(4)); (X(3), -- , X(3463));
(X(4), X(25), X(4)); (X(7), X(1617), X(279)); (X(8), -- , X(280)); ...
Curious case: NOTE THAT if P=X(7) then Q=X(1617) = TCC-PERSPECTOR OF X(57) and T=X(279)=X(57)-cross conjugate of X(7).
Are there more points P satisfying this condition?
That is, if P is a triangle center, is there a triangle center W verifying T=W-cross conjugate of P and Q=TCC-perspector of W?