IGO 2020 E2

A parallelogram $ABCD$ is given ( $AB \neq BC$ ). Points $E$ and $G$ are chosen on the line $\overline{CD}$ such that $\overline{AC}$ is the angle bisector of both angles $\angle EAD$ and $\angle BAG$ . The line $\overline{BC}$ intersects $\overline{AE}$ and $\overline{AG}$ at $F$ and $H$ , respectively. Prove that the line $\overline{FG}$ passes through the midpoint of $HE$ . Proposed by Mahdi Etesamifard