Question 25
Let $A A^{\prime} B C C^{\prime} B^{\prime}$ be a convex cyclic hexagon such that $A C$ is tangent to the incircle of the triangle $A^{\prime} B^{\prime} C^{\prime}$, and $A^{\prime} C^{\prime}$ is tangent to the incircle of the triangle $A B C$. Let the lines $A B$ and $A^{\prime} B^{\prime}$ meet at $X$ and let the lines $B C$ and $B^{\prime} C^{\prime}$ meet at $Y$.
Prove that if $X B Y B^{\prime}$ is a convex quadrilateral, then it has an incircle.
Prove that if $X B Y B^{\prime}$ is a convex quadrilateral, then it has an incircle.