MEMO 2019 I3

Let $ABC$ be an acute-angled triangle with $AC>BC$ and circumcircle $\omega$ . Suppose that $P$ is a point on $\omega$ such that $AP=AC$ and that $P$ is an interior point on the shorter arc $BC$ of $\omega$ . Let $Q$ be the intersection point of the lines $AP$ and $BC$ . Furthermore, suppose that $R$ is a point on $\omega$ such that $QA=QR$ and $R$ is an interior point of the shorter arc $AC$ of $\omega$ . Finally, let $S$ be the point of intersection of the line $BC$ with the perpendicular bisector of the side $AB$ . Prove that the points $P, Q, R$ and $S$ are concyclic.

Proposed by Patrik Bak, Slovakia