Reflections of a line through H in parallels to altitudes through A,B,C

Let ABC be a triangle, HaHbHc the orthic triangle, L a line through H and Lab,Lac the parallels through A to altitudes BHb, CHc, resp. The reflections of L in Lab, Lac intersect at A'. Similarly B',C'.

The triangles ABC, A'B'C' are perspective. The perspector lies on the circumcircle of ABC, where concur the reflections of L in the sidelines of ABC.

Note: The triangle A'B'C' is also perspective with the antimedial and medial triangles of ABC.

APH, 25 November 2011