Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle BAP = \angle CAQ$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcentre of triangle $BCR$. Prove that $AO \perp PQ$.