Question 50
Let $M$ be a point on the side $BC$ of a right triangle $ABC$ with the right angle at $A$. Denote the midpoint of the segment $AM$ by $P$. Let $P_{B}$ and $P_{C}$ be the reflections of the point $P$ across the sides $AC$ and $AB$, and let $B'$ and $C'$ be the reflections of the points $B$ and $C$ across the point $P$. Suppose that the points $B'$, $C'$, $P_{B}$ and $P_{C}$ are collinear. Prove that then $M$ is the midpoint of the side $BC$.