Question 37
The incircle of the acute triangle $ABC$ touches the segments $BC$, $CA$, $AB$ at points $D$, $E$, $F$ respectively. Let $S$ be the incenter and $P$ be the intersection of the line $DS$ and the segment $EF$. If $M$ is the midpoint of the segment $BC$ prove that the points $A$, $P$ and $M$ are collinear.