ELMO 2012 Problem 1

In acute triangle $ABC$ , let $D,E,F$ denote the feet of the altitudes from $A,B,C$ , respectively, and let $\omega$ be the circumcircle of $\triangle AEF$ . Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$ , respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$ .

Ray Li.