Question 48
Let $A$, $B$, $C$ be colinear points in this order, $\mathcal{C}$ an arbitrary circle passing through $B$ and $C$, and $l$ an arbitrary line different from $BC$, passing through $A$ and intersecting $\mathcal{C}$ at $M$ and $N$. The bisectors of the angles $\angle C M B$ and $\angle C N B$ intersect $BC$ at $P$ and $Q$. Prove that $AP \cdot AQ = AB \cdot AC$.