KöMaL 2026 C1904

$AB$ is the hypotenuse of right triangle $ABC$. Let $D$ be the foot of the altitude from $C$. Let $E$ be the foot of the perpendicular from $D$ to $AC$, and let $F$ be the foot of the perpendicular from $E$ to $AB$. Given that $F$ is the midpoint of the hypotenuse, find the lengths of the legs of the triangle.