A Generalization of IMO 2008 Problem 1

Let ABC be a triangle. Let P be a point and Q the isogonal conjugate of P with respect to ABC. Let R, S, and T be the orthogonal projections of P onto AB, AC, and BC. Let the circles with centers at R, S, and T and passing through Q intersect the sidelines of BC, AC, and AB at A', A'', B', B'', C', and C''. Then A', A'', B', B'', C', and C'' are concyclic.