TSTST 2019 Problem 2

Let $A B C$ be an acute triangle with circumcircle $\Omega$ and orthocenter $H$. Points $D$ and $E$ lie on segments $A B$ and $A C$ respectively, such that $A D=A E$. The lines through $B$ and $C$ parallel to $\overline{D E}$ intersect $\Omega$ again at $P$ and $Q$, respectively. Denote by $\omega$ the circumcircle of $\triangle A D E$. (a) Show that lines $P E$ and $Q D$ meet on $\omega$. (b) Prove that if $\omega$ passes through $H$, then lines $P D$ and $Q E$ meet on $\omega$ as well.