RU-SPB 2016 Problem 96

Incircle of $\triangle ABC$ touch $AC$ at $D$ . $BD$ intersect incircle at $E$ . Points $F,G$ on incircle are such points, that $FE \parallel BC,GE \parallel AB$ . $I_1,I_2$ are incenters of $DEF,DEG$ . Prove that angle bisector of $\angle GDF$ passes though the midpoint of $I_1I_2$ .