2020 IMO Shortlist G5

Let $A B C D$ be a cyclic quadrilateral with no two sides parallel. Let $K, L, M$, and $N$ be points lying on sides $A B, B C, C D$, and $D A$, respectively, such that $K L M N$ is a rhombus with $K L \| A C$ and $L M \| B D$. Let $\omega_{1}, \omega_{2}, \omega_{3}$, and $\omega_{4}$ be the incircles of triangles $A N K$, $B K L, C L M$, and $D M N$, respectively. Prove that the internal common tangents to $\omega_{1}$ and $\omega_{3}$ and the internal common tangents to $\omega_{2}$ and $\omega_{4}$ are concurrent. (Poland)

设 $A B C D$ 为没有两条边平行的循环四边形。设 $K、L、M$ 和 $N$ 分别为边 $A B、B C、C D$ 和 $D A$ 上的点，使得 $K L M N$ 是菱形，$K L \| A C$ 和 $L M \| B D$。设$\omega_{1}、\omega_{2}、\omega_{3}$和$\omega_{4}$分别为三角形$A N K$、$B K L、C L M$和$D M N$的内切圆。证明 $\omega_{1}$ 和 $\omega_{3}$ 的内公切线与 $\omega_{2}$ 和 $\omega_{4}$ 的内公切线是并发的。 (波兰)