ELMO SL 2012 G6

In $\triangle ABC$ , $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$ , respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$ . $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$ , and $R_1S_1\cap R_2S_2=Y$ . Prove that $X,Y,H$ are collinear.

Ray Li.