IGO 2022 A4

Let $ABCD$ be a trapezoid with $AB\parallel CD$ . Its diagonals intersect at a point $P$ . The line passing through $P$ parallel to $AB$ intersects $AD$ and $BC$ at $Q$ and $R$ , respectively. Exterior angle bisectors of angles $DBA$ , $DCA$ intersect at $X$ . Let $S$ be the foot of $X$ onto $BC$ . Prove that if quadrilaterals $ABPQ$ , $CDQP$ are circumcribed, then $PR=PS$ .

Proposed by Dominik Burek, Poland