ELMO SL 2014 G6

Let $ABCD$ be a cyclic quadrilateral with center $O$ . Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$ , while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$ . Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$ . Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$ . Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$ .

Proposed by Yang Liu