BALTIC-WAY 2020 Problem 13

Let $A B C$ be an acute triangle with circumcircle $\omega$. Let $\ell$ be the tangent line to $\omega$ at $A$. Let $X$ and $Y$ be the projections of $B$ onto lines $\ell$ and $A C$, respectively. Let $H$ be the orthocenter of $B X Y$. Let $C H$ intersect $\ell$ at $D$. Prove that $B A$ bisects angle $C B D$.