Question 155
Let $ABC$ be an acute triangle with circumcircle $\Omega$ and orthocenter $H$. Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively, such that $AD = AE$. The lines through $B$ and $C$ parallel to $DE$ intersect $\Omega$ again at $P$ and $Q$, respectively. Denote by $\omega$ the circumcircle of $\triangle ADE$.
a. Show that lines $PE$ and $QD$ meet on $\omega$.
b. Prove that if $\omega$ passes through $H$, then lines $PD$ and $QE$ meet on $\omega$ as well.
a. Show that lines $PE$ and $QD$ meet on $\omega$.
b. Prove that if $\omega$ passes through $H$, then lines $PD$ and $QE$ meet on $\omega$ as well.