Question 130
Let $ABC$ be an acute triangle with $AB \ne AC$ and $H$ its orthocenter. Consider a point $D$ on the side $BC$. The circumcircles of triangles $ABD$ and $ACD$ meet again $AC$ and $AB$ in $E$ and $F$, respectively. Lines $BE$ and $CF$ meet in point $P$. Prove that $HP$ is parallel to $BC$ if and only if the line $AD$ contains the circumcenter of $ABC$.