IrMO 2016 Problem 4

Let $ABC$ be a triangle with $|AC| \ne |BC|$ . Let $P$ and $Q$ be the intersection points of the line $AB$ with the internal and external angle bisectors at $C$ , so that $P$ is between $A$ and $B$ . Prove that if $M$ is any point on the circle with diameter $PQ$ , then $\angle AMP = \angle BMP$ .