Let ABC be a triangle, and let C′ and A′ be the feet of its altitudes issuing from the vertices C and A, respectively. Denote by P the midpoint of the segment C′A′. The circumcircles of triangles AC′P and CA′P have a common point apart from P; denote this common point by Q. Prove that:
(a) The point Q lies on the circumcircle of the triangle ABC.
(b) The line PQ passes through the point B.
(c) We have CQAQ=CBAB.
Darij geometrycircumcircletrigonometrygeometry proposed