Question 166
Let $ABC$ be an acute-angled triangle with $AC \lt BC$. A circle passes through $A$ and $B$, and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$, respectively. The circumcircles of the triangles $ABC$ and $A_1B_1C$ meet again at $P$. The segments $AB_1$ and $BA_1$ cross at $S$. Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$, respectively. Prove that the points $P$, $Q$, $R$, and $C$ are concyclic.