OMCC 2016 Problem 2

Let $ABC$ be an acute-angled triangle, $\Gamma$ its circumcircle and $M$ the midpoint of $BC$ . Let $N$ be a point in the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle NAC= \angle BAM$ . Let $R$ be the midpoint of $AM$ , $S$ the midpoint of $AN$ and $T$ the foot of the altitude through $A$ . Prove that $R$ , $S$ and $T$ are collinear.