A semicircle with center $O$ and diameter $AB$ is given. Point $M$ on the extension of $AB$ is taken so that $AM > BM$. A line through $M$ intersects the semicircle at $C$ and $D$ so that $CM < DM$. The circumcircles of triangles $AOD$ and $OBC$ meet again at point $K$. Prove that $OK$ and $KM$ are perpendicular