2022 IMO Shortlist G3

Let $A B C D$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $A C$ is tangent to the circle $A D Q$, and the line $B D$ is tangent to the circle $B C P$. Let $M$ and $N$ be the midpoints of $B C$ and $A D$, respectively. Prove that the following three lines are concurrent: line $C D$, the tangent of circle $A N Q$ at point $A$, and the tangent to circle $B M P$ at point $B$. (Slovakia)

设 $A B C D$ 为循环四边形。假设点$Q、A、B、P$ 依次共线，即直线$A C$ 与圆$A D Q$ 相切，直线$B D$ 与圆$B C P$ 相切。设$M$和$N$分别为$B C$和$A D$的中点。证明以下三条直线是并发的：直线$C D$、圆$A N Q$ 在点$A$ 的切线以及圆$B M P$ 在点$B$ 的切线。 (斯洛伐克)