2022 IMO Shortlist G8

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.

令 $A A^{\prime} B C C^{\prime} B^{\prime}$ 为凸循环六边形，使得 $A C$ 与三角形 $A^{\prime} B^{\prime} C^{\prime}$ 的内圆相切，并且 $A^{\prime} C^{\prime}$ 与三角形 $A B C$ 的内圆相切。让$A B$ 和$A^{\prime} B^{\prime}$ 线在$X$ 处相交，并让$B C$ 和$B^{\prime} C^{\prime}$ 在$Y$ 处相交。证明如果 $X B Y B^{\prime}$ 是凸四边形，则它有内切圆。