Question 92
Let $H$ be the orthocenter of $\triangle ABC$. The points $A_1 \neq A$, $B_1 \neq B$ and $C_1 \neq C$ lie respectively on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$, and $A_1H = B_1H = C_1H$. Denote by $H_1$, $H_2$ and $H_3$ the orthocenters of $\triangle A_1BC$, $\triangle B_1CA$ and $\triangle C_1AB$, respectively. Prove that $\triangle A_1B_1C_1$ and $\triangle H_1H_2H_3$ have the same orthocenter.