ELMO 2023 Problem 4

Let $ABC$ be an acute scalene triangle with orthocenter $H$. Line $BH$ intersects $\overline{AC}$ at $E$ and line $CH$ intersects $\overline{AB}$ at $F$. Let $X$ be the foot of the perpendicular from $H$ to the line through $A$ parallel to $\overline{EF}$. Point $B_1$ lies on line $XF$ such that $\overline{BB_1}$ is parallel to $\overline{AC}$, and point $C_1$ lies on line $XE$ such that $\overline{CC_1}$ is parallel to $\overline{AB}$. Prove that points $B$, $C$, $B_1$, $C_1$ are concyclic.

Proposed by Luke Robitaille