ELMO SL 2012 G4

Circles $\Omega$ and $\omega$ are internally tangent at point $C$ . Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$ , where $E$ is the midpoint of $AB$ . Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$ . If $M$ is the midpoint of major arc $AB$ , show that $\tan \angle ZEP = \tfrac{PE}{CM}$ .

Ray Li.