Question 16
Given an acute triangle $ABC$ in which $|AB| \gt |BC| \gt |AC|$. Let $D$ be a point on the segment $BC$, different from $C$, such that $|AC| = |AD|$. Denote by $H$ the orthocentre of the triangle $ABC$ and by $A_{1}$ and $B_{1}$ the feet of the altitudes from $A$ and $B$. The line $DH$ meets the line $AC$ at the point $E$ and the line $A_{1}B_{1}$ at the point $F$. Let $G$ be the intersection of the lines $AF$ and $BH$. Prove that the triangles $HBD$ and $HGE$ are similar.