ELMO SL 2014 G2

$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$ . Let $AB \cap CD = E$ , $AD \cap BC = F$ . Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$ , respectively. Let $\omega \cap \omega_1 = G$ , $\omega \cap \omega_2 = H$ . Show that $AC, BD, GH$ are concurrent.

Proposed by Yang Liu