Question 38
Let $ABC$ be an acute triangle. The circle centered at $A$ tangent to side $BC$ meets side $AB$ at $B_1$ and side $CA$ at $C_2$. The circle centered at $B$ tangent to side $CA$ meets side $BC$ at $C_1$ and side $AB$ at $A_2$. The circle centered at $C$ tangent to side $AB$ meets side $CA$ at $A_1$ and side $BC$ at $B_2$. Prove that the triangle bounded by the lines $A_1A_2$, $B_1B_2$, and $C_1C_2$ is similar to $ABC$.