SHARYGIN 2012 Problem 12

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ . A line passing through $O$ and parallel to $BC$ meets $AB$ and $AC$ in points $P$ and $Q$ respectively. The sum of distances from $O$ to $AB$ and $AC$ is equal to $OA$ . Prove that $PB + QC = PQ$ .