IGO 2022 A5

Let $ABC$ be an acute triangle inscribed in a circle $\omega$ with center $O$ . Points $E$ , $F$ lie on its side $AC$ , $AB$ , respectively, such that $O$ lies on $EF$ and $BCEF$ is cyclic. Let $R$ , $S$ be the intersections of $EF$ with the shorter arcs $AB$ , $AC$ of $\omega$ , respectively. Suppose $K$ , $L$ are the reflection of $R$ about $C$ and the reflection of $S$ about $B$ , respectively. Suppose that points $P$ and $Q$ lie on the lines $BS$ and $RC$ , respectively, such that $PK$ and $QL$ are perpendicular to $BC$ . Prove that the circle with center $P$ and radius $PK$ is tangent to the circumcircle of $RCE$ if and only if the circle with center $Q$ and radius $QL$ is tangent to the circumcircle of $BFS$ .

Proposed by Mehran Talaei