CZ-SK-MO 2022 Problem 4

Let $ABCD$ be a convex quadrilateral with $AB = BC = CD$ and $P$ its intersection of diagonals. Denote by $O_1$ , $O_2$ the circumcenters of triangles $ABP$ , $CDP$ , respectively. Prove that $O_1BCO_2$ is a parallelogram.

(Patrik Bak)