The construction can be generalized. Let P=(u:v:w) be a point in the plane of triangle ABC, and let DEF be its cevian triangle. Circle (PEF) intersects line AB at Ac and AC at Ab. Define Ba, Bc, Cb, Ca cyclically. The six points lie on a conic. The equation of the conic is not nice, but the center of the conic has a simple form
( a^2 (u+v)(u+w) v w : b^2 (v+w)(v+u) u w : : c^2 (u+w)(v+w) u v ).
It is the cevapoint of X(6) and the isogonal conjugate of P.
The construction can be generalized. Let P=(u:v:w) be a point in the plane of triangle ABC, and let DEF be its cevian triangle. Circle (PEF) intersects line AB at Ac and AC at Ab. Define Ba, Bc, Cb, Ca cyclically. The six points lie on a conic. The equation of the conic is not nice, but the center of the conic has a simple form
ReplyDelete( a^2 (u+v)(u+w) v w : b^2 (v+w)(v+u) u w : : c^2 (u+w)(v+w) u v ).
It is the cevapoint of X(6) and the isogonal conjugate of P.
Elias M. Hagos