2020 OIM Problems/Problem 6

Problem

Let $ABC$ be an acute and scalene triangle. Let $H$ be the orthocenter and $O$ the circumcenter of triangle $ABC$ , and let $P$ be an interior point of the segment $HO$ . The circumference with center $P$ and radius $PA$ again intersects the lines $AB$ and $AC$ at points $R$ and $S$ , respectively. We denote by $Q$ the point symmetrical to point $P$ with respect to the bisector of $BC$ . Prove that the points $P$ , $Q$ , $R$ and $S$ belong to the same circle.