Question 100
Suppose $O$ is the circumcenter of $\triangle ABC$ and $E$, $F$ are points on segments $CA$ and $AB$ respectively with $E, F \ne A$. Let $P$ be a point such that $PB = PF$ and $PC = PE$. Let $OP$ intersect $CA$ and $AB$ at points $Q$ and $R$ respectively. Let the line passing through $P$ and perpendicular to $EF$ intersect $CA$ and $AB$ at points $S$ and $T$ respectively. Prove that points $Q$, $R$, $S$, and $T$ are concyclic.