Question 9
Let $APD$ be an acute-angled triangle and let $B \in (AP)$, $C \in (PD)$ be two points. The diagonals of the quadrilateral $ABCD$ meet at the point $Q$. Denote by $H_1$ and $H_2$ the orthocenters of the triangles $APD$ and $BPC$ respectively. The circumcircles of the triangles $ABQ$ and $CDQ$ meet again at the point $X$ ($X \neq Q$) and the circumcircles of the triangles $ADQ$ and $BCQ$ meet again at the point $Y$ ($Y \neq Q$).
Show that if the line $H_1H_2$ passes through the point $X$, then it also passes through the point $Y$.
Show that if the line $H_1H_2$ passes through the point $X$, then it also passes through the point $Y$.