KMO 2012 Problem 6

Let $w$ be the incircle of triangle $ABC$ . Segments $BC, CA$ meet with $w$ at points $D, E$ . A line passing through $B$ and parallel to $DE$ meets $w$ at $F$ and $G$ . ( $F$ is nearer to $B$ than $G$ .) Line $CG$ meets $w$ at $H ( \ne G )$ . A line passing through $G$ and parallel to $EH$ meets with line $AC$ at $I$ . Line $IF$ meets with circle $w$ at $J (\ne F )$ . Lines $CJ$ and $EG$ meets at $K$ . Let $l$ be the line passing through $K$ and parallel to $JD$ . Prove that $l, IF, ED$ meet at one point.