MEMO 2023 T5

We are given a convex quadrilateral $ABCD$ whose angles are not right. Assume there are points $P, Q, R, S$ on its sides $AB, BC, CD, DA$, respectively, such that $PS \parallel BD, SQ \perp BC, PR \perp CD$. Furthermore, assume that the lines $PR, SQ,$ and $AC$ are concurrent. Prove that the points $P, Q, R, S$ are concyclic.