BALTIC-WAY 2018 Problem 12

The altitudes $B B_{1}$ and $C C_{1}$ of an acute triangle $A B C$ intersect in point $H$. Let $B_{2}$ and $C_{2}$ be points on the segments $B H$ and $C H$, respectively, such that $B B_{2}=B_{1} H$ and $C C_{2}=C_{1} H$. The circumcircle of the triangle $B_{2} H C_{2}$ intersects the circumcircle of the triangle $A B C$ in points $D$ and $E$. Prove that the triangle $D E H$ is right-angled.