ELMO SL 2014 G4

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ . Define $E = AA \cap CD$ , $F = AA \cap BC$ , $G = BE \cap \omega$ , $H = BE \cap AD$ , $I = DF \cap \omega$ , and $J = DF \cap AB$ . Prove that $GI$ , $HJ$ , and the $B$ -symmedian are concurrent.

Proposed by Robin Park