Question 1
Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $\overline{BC}$ and passing through $H$ intersects the sideline $BC$ at points $A_1$ and $A_2$. Similarly, the circle centred at the midpoint of $CA$ and passing through $H$ meets line $CA$ at $B_1,B_2$, and the circle centred at the midpoint of $AB$ and passing through $H$ meets line $AB$ at $C_1,C_2$. Prove that six points $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ are concyclic.