Problem

Source: China south east mathematical olympiad 2006 day2 problem 5

Tags: geometry, trigonometry, symmetry, geometry unsolved



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 $FG=\frac{BC}{2}$.