Question 190
Let $ABC$ be a triangle with circumcircle $\Gamma$, whose incircle touches $BC$, $CA$, $AB$ at $D$, $E$, $F$. We draw a circle tangent to segment $BC$ at $D$ and to minor $\operatorname{arc} \widehat{BC}$ of $\Gamma$ at the point $A_{1}$. Define $B_{1}$ and $C_{1}$ in a similar way. Prove that lines $A_{1}D$, $B_{1}E$, $C_{1}F$ are concurrent.