VMO 2009 Problem 3

Let $A$ , $B$ be two fixed points and $C$ is a variable point on the plane such that $\angle ACB=\alpha$ (constant) ( $0^{\circ}\le \alpha\le 180^{\circ}$ ). Let $D$ , $E$ , $F$ be the projections of the incenter $I$ of triangle $ABC$ to its sides $BC$ , $CA$ , $AB$ , respectively. Denoted by $M$ , $N$ the intersections of $AI$ , $BI$ with $EF$ , respectively. Prove that the length of the segment $MN$ is constant and the circumcircle of triangle $DMN$ always passes through a fixed point.