CWMO 2015 Problem 2

Two circles $\left(\Omega_1\right),\left(\Omega_2\right)$ touch internally on the point $T$ . Let $M,N$ be two points on the circle $\left(\Omega_1\right)$ which are different from $T$ and $A,B,C,D$ be four points on $\left(\Omega_2\right)$ such that the chords $AB, CD$ pass through $M,N$ , respectively. Prove that if $AC,BD,MN$ have a common point $K$ , then $TK$ is the angle bisector of $\angle MTN$ .

• $\left(\Omega_2\right)$ is bigger than $\left(\Omega_1\right)$