Question 31
Let $\triangle ABC$ be an acute triangle such that $|AB| \gt |AC|$. 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. The line $DH$ intersects the line $AC$ at $E$ and the line $A_1B_1$ at $F$. Let $G$ be the intersection of the lines $AF$ and $BH$. Prove that $EG, CH$ and $AD$ meet at the same point.