KöMaL 2007 B4036

The circles $k_1$ and $k_2$ touch each other on the outside at $P$. One of their common tangents not passing through $P$ is $e$, which touches $k_1$ and $k_2$ at $A$ and $B$, respectively. The line $f$ is parallel to $e$, touches $k_1$ at $C$ and intersects $k_2$ at $D$ and $E$. Prove that the common chord of the circumcircles of the triangles ABC and ADE contains $P$.