BALKAN 2025 Problem 2

Let $\triangle ABC$ be an acute- angled triangle with orthocentre $H$ and let $D$ be an arbitrary interior point on side $BC$ . Suppose $E$ and $F$ are points on the segments $AB$ and $AC$ respectively such that the quadrilaterals $ABDF$ and $ACDE$ are cyclic, and let $BF$ and $CE$ intersect at $P$ . Let $L$ be the point of line $HA$ such that $LC$ is tangent to the circumcircle of triangle $\triangle PBC$ at point $C$ . Let lines $BH$ and $CP$ intersect at $X$ . Prove that $D, L$ and $X$ are collinear.