2014 IMO Shortlist G1

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 $A B C$. (Georgia)

点$P$和$Q$选在锐角三角形$A B C$的边$B C$上，使得$\angle P A B=\angle A C B$和$\angle Q A C=\angle C B A$。点$M$和$N$分别取在射线$A P$和$A Q$上，因此$A P=P M$和$A Q=Q N$。证明直线 $B M$ 和 $C N$ 相交于三角形 $A B C$ 的外接圆上。 (乔治亚州)