Let ABC be a triangle, P a point, A'B'C' the pedal triangle of P, A"B"C" its antipodal triangle (in the pedal circle) and A*B*C* the circumcevian triangle of P.
Which is the locus of P such that A*B*C*, A"B"C" are perspective?
Antreas P. Hatzipolakis, Hyacinthos #21738
A*B*C*, A"B"C" are perspective if P is on the circumcircle (the circumcevian triangle degenerates) or on the K003="McCay cubic" or on K191="circumcircle pedal cubic".
If P is on K191="circumcircle pedal cubic", the perspector of the triangles A*B*C* and A"B"C" is a point in the circumcircle.
Angel Montesdeoca, Hyacinthos #21743
The cubic is S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0 (not K191) and will be K634 in Bernard Gibert's list.