Question 100
Let $A, B, C, D$ be four points lying in this order on a circle $\omega$. Let $P$ and $Q$ be two points on the line $(A B)$ such that the points $Q, A, B, P$ are collinear in this order, the circumcircle of $A D Q$ is tangent to the line $(A C)$, and the circumcircle of $B C P$ is tangent to the line $(B D)$. Let $M$ and $N$ be the midpoints of the segments $[B C]$ and $[A D]$ respectively.
Prove that the tangent at $A$ to the circumcircle of $A N Q$, the tangent at $B$ to the circumcircle of $B M P$ and the line $(C D)$ are concurrent.
Prove that the tangent at $A$ to the circumcircle of $A N Q$, the tangent at $B$ to the circumcircle of $B M P$ and the line $(C D)$ are concurrent.