ÖMO 2015 Problem 2

We are given a triangle $ABC$ . Let $M$ be the mid-point of its side $AB$ .

Let $P$ be an interior point of the triangle. We let $Q$ denote the point symmetric to $P$ with respect to $M$ .

Furthermore, let $D$ and $E$ be the common points of $AP$ and $BP$ with sides $BC$ and $AC$ , respectively.

Prove that points $A$ , $B$ , $D$ , and $E$ lie on a common circle if and only if $\angle ACP = \angle QCB$ holds.

(Karl Czakler)