Question 32
Let $ABC$ be an acute triangle such that $|AB| \gt |BC| \gt |AC|$. Let $D$ be a point different from $C$ on the segment $BC$, such that $|AC| = |AD|$. Let $H$ denote the orthocentre of the triangle $ABC$, and let $A_1, B_1$ be the feet of the altitudes from $A$ and $B$, respectively. Denote the intersection of the lines $A_1B_1$ and $DH$ by $E$. Prove that $B, D, B_1$ and $E$ are concyclic.