MathNet 2011 Question 18

The incircle of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let $G$ be a point on the incircle such that $FG$ is a diameter. The lines $EG$ and $FD$ intersect at $H$. Prove that $CH \parallel AB$.