Question 31
The circles $K_1$ and $K_2$ intersect in two distinct points $A$ and $B$. The tangents to $K_1$ through $A$ and $B$ intersect at $T$. Let $M$ be an arbitrary point on the circle $K_1$ distinct from $A$ and $B$. The line $MT$ meets the circle $K_1$ again at $C$, the line $MA$ meets the circle $K_2$ again at $K$ and the line $AC$ meets the circle $K_2$ again at $L$. Prove that the line $MC$ contains the midpoint of the segment $KL$.