Let $AB$ be a chord of a circle $\Gamma$. Let $\omega$ be a circle tangent to chord $AB$ at $K$ and internally tangent to $\omega$ at $T$. Then ray $TK$ passes through the midpoint $M$ of the arc $AB$ not containing $T$.
** This question might be asked before. If them don't down vote. Just close it marking duplicate and send me the link of the question. I search out but failed to find **.
I tried my best by doing angle chasing but failed.I think radical axis will not work as well. If anyone have any pure geometry answer please post. I think inversion will work. But i know inversion a little. But I guess Inversion around $M$ will work. Please help me.
Hint:
Homothety in $T$ which takes smaller circle to bigger will do. It takes $K$ to $M$ and line $AB$ to parallel line through $M$ since $K$ is on $AB$. Thus this new line is also tangent to bigger circle, since $AB$ is tangent to smaller. Now it should not be difficult to see that $AMB$ is isosceles (remember tangent-chord property).