The Hagge Circle

Let ABC be a triangle and H its orthocenter. Let P be a point and AP, BP, and CP meet the circumcircle of ABC again at A', B', and C'. Let A'', B'', and C'' be the reflections of A', B', and C' about BC, AC, and AB. Let A''P, B''P, and C''P intersect AH, BH, and CH at X, Y, and Z. Then H, A'', B'', C'', X, Y, and Z are concyclic.