Let $ABC$ be an acute triangle with altitudes $AD,BE$ and $CF$. Denote $\omega_1,\omega_2$ the circumcircles of $\triangle AEB, \triangle AFC$, respectively. Suppose the line through $A$ parallel to $EF$ intersects $\omega_1$ and $\omega_2$ at $P$ and $Q$, respectively. Show that the circumcenter of $\triangle PQD$ lies on $AD$