Question 72
Let $ABCD$ be a tangential quadrilateral. Let $\omega$ be externally inscribed circle in $ABCD$, tangent to $AD$, $BC$ and $AB$. Denote by $X_{AB}$ the point of tangency of $\omega$ and the circle that passes through $A$ and $B$ and internally tangent to $\omega$. Let us define $X_{BC}$, $X_{CD}$ and $X_{DA}$, analogously. Prove that the bisectors of the angles $\angle AX_{AB}B$, $\angle BX_{BC}C$, $\angle CX_{CD}D$ and $\angle DX_{DA}A$ are concurrent.