TSTST 2018 Problem 5

Let $A B C$ be an acute triangle with circumcircle $\omega$, and let $H$ be the foot of the altitude from $A$ to $\overline{B C}$. Let $P$ and $Q$ be the points on $\omega$ with $P A=P H$ and $Q A=Q H$. The tangent to $\omega$ at $P$ intersects lines $A C$ and $A B$ at $E_{1}$ and $F_{1}$ respectively; the tangent to $\omega$ at $Q$ intersects lines $A C$ and $A B$ at $E_{2}$ and $F_{2}$ respectively. Show that the circumcircles of $\triangle A E_{1} F_{1}$ and $\triangle A E_{2} F_{2}$ are congruent, and the line through their centers is parallel to the tangent to $\omega$ at $A$.