Question 87
Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers at $C_1$ and $C_2$, respectively, intersect at two distinct points $A$ and $B$. Points $P$ and $Q$ are varying points on $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively, such that $P, Q$ and $B$ are collinear and $B$ is always between $P$ and $Q$. Let lines $P C_1$ and $Q C_2$ intersect at $R$, let $I$ be the incenter of $\triangle P Q R$, and let $S$ be the circumcenter of $\triangle P I Q$. Show that as $P$ and $Q$ vary, $S$ traces an arc of a circle whose center is concyclic with $A, C_1$ and $C_2$.