Let P and $Q$} be on segment BC of an acute triangle ABC such that \angle PAB=\angle BCA and \angle CAQ=\angle ABC. Let M and N be the points on AP and AQ, respectively, such that P is the midpoint of AM and Q is the midpoint of AN.

Prove that the intersection D of BM and CN is on the circumference of triangle ABC.

The solution can be found in a separate file.