BALTIC-WAY 2024 Problem 13

Let $A B C$ be an acute triangle with orthocentre $H$. Let $D$ be a point outside the circumcircle of triangle $A B C$ such that $\angle A B D=\angle D C A$. The reflection of $A B$ in $B D$ intersects $C D$ at $X$. The reflection of $A C$ in $C D$ intersects $B D$ at $Y$. The lines through $X$ and $Y$ perpendicular to $A C$ and $A B$, respectively, intersect at $P$. Prove that points $D, P$ and $H$ are collinear.