Question 30
Let $ABC$ be a triangle with $A'$, $B'$, $C'$ as the midpoints of $BC$, $CA$, $AB$ respectively. The circle $(\omega_A)$ of center $A$ with sufficiently large radius cuts $B'C'$ at $X_1$, $X_2$. Define circles $(\omega_B)$, $(\omega_C)$ with $Y_1$, $Y_2$, $Z_1$, $Z_2$ similarly. Suppose that these circles have the same radius. Prove that $X_1$, $X_2$, $Y_1$, $Y_2$, $Z_1$, $Z_2$ are concyclic.