Concurrence On The Nine-Point Circle (No proven yet)

Much more interesting than the particular case I proposed, is the generalization by Francisco Javier García Capitán at ADGEOM1280.

Given a triangle ABC and a point P. Call A1B1C1 be the circumcevian triangle of the complement of P with respect the medial triangle. Call A2B2C2 the triangle formed by the reflections of the vertices of the pedal triangle of P on the center of the pedal circle of P, that is the antipodes of the vertices of the pedal triangle on its circumcircle. The triangles A1B1C1 and A2B2C2 are perspective at a point Q on the nine point circle.