Question 29
Given an acute-angled triangle $ABC$ with an altitude $AD$ and orthocenter $H$. Let $E$ be the reflection of $H$ with respect to $A$. A point $X$ lies on the circumcircle of the triangle $BDE$ such that $DX \parallel AC$, and similarly a point $Y$ lies on the circumcircle of the triangle $CDE$ such that $DY \parallel AB$. Prove that the circumcircles of triangles $AXY$ and $ABC$ are tangent to each other.