KöMaL 2021 C1663

The circles $k_1$ and $k_2$ touch each other externally at point $E$. Lines $f$ and $g$ pass through point $E$. One of the common external tangents touches the circle $k_1$ and $k_2$ at points $C$ and $D$, respectively. Line $h$ is obtained by dropping perpendiculars from point $C$ onto lines $f$ and $g$, and connecting the feet of the perpendiculars. Line $m$ is obtained in the same way, with perpendiculars from point $D$. Prove that $h$ and $m$ are perpendicular to each other.