KöMaL 2024 B5379

Let $ABC$ be a triangle with a right angle at $C$. On the hypotenuse $AB$, let $H$ and $D$ be the feet of the altitude and the angle bisector starting from $C$, respectively. The bisector of angle $AHC$ meets $AC$ at point $E$, and the bisector of angle $CHB$ meets $BC$ at $F$. Mark a point $M$ on line segment $HE$ and a point $N$ on line segment $HF$ that satisfy $HM:HE=HN:HF$. Show that lines $CD$, $AM$ and $BN$ are concurrent.