CN-SMO 2006 Problem 5

In $\triangle ABC$ , $\angle A=60^\circ$ . $\odot I$ is the incircle of $\triangle ABC$ . $\odot I$ is tangent to sides $AB$ , $AC$ at $D$ , $E$ , respectively. Line $DE$ intersects line $BI$ and $CI$ at $F$ , $G$ respectively. Prove that $FG=\frac{BC}{2}$ .