SHARYGIN 2022 Problem 96

Lateral sidelines $AB$ and $CD$ of a trapezoid $ABCD$ ( $AD >BC$ ) meet at point $P$ . Let $Q$ be a point of segment $AD$ such that $BQ = CQ$ . Prove that the line passing through the circumcenters of triangles $AQC$ and $BQD$ is perpendicular to $PQ$ .