RMO 2010 Problem 5

Let $ABC$ be a triangle in which $\angle A = 60^\circ$ . Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$ . Let $M$ be the reflection of $A$ in line $EF$ . Prove that $M$ lies on $BC$ .