JBMO-SL 2016 G7

Let $[A B]$ be a chord of a circle $(c)$ centered at $O$, and let $K$ be a point on the segment $(A B)$ such that $A K<B K$. Two circles through $K$, internally tangent to (c) at $A$ and $B$, respectively, meet again at $L$. Let $P$ be one of the points of intersection of the line $K L$ and the circle (c), and let the lines $A B$ and $L O$ meet at $M$. Prove that the line $M P$ is tangent to the circle $(c)$.