IT-CESENATICO 2013 Problem 2

In triangle $ABC$ , suppose we have $a> b$ , where $a=BC$ and $b=AC$ . Let $M$ be the midpoint of $AB$ , and $\alpha, \beta$ are inscircles of the triangles $ACM$ and $BCM$ respectively. Let then $A'$ and $B'$ be the points of tangency of $\alpha$ and $\beta$ on $CM$ . Prove that $A'B'=\frac{a - b}{2}$ .