Question 58
The points $P$ and $Q$ are chosen on the side $B C$ of an acute-angled triangle $A B C$ so that $\angle P A B = \angle A C B$ and $\angle Q A C = \angle C B A$. The points $M$ and $N$ are taken on the rays $A P$ and $A Q$, respectively, so that $A P = P M$ and $A Q = Q N$. Prove that the lines $B M$ and $C N$ intersect on the circumcircle of the triangle $A B C$.