KöMaL 2025 B5484

Let $M$ denote the orthocenter of acute triangle $ABC$, and let $F$ denote the midpoint of side $BC$. Let $T$ denote the foot of the perpendicular from $M$ to the inner angle bisector at vertex $A$. Line $FT$ intersects side $AC$ at point $D$. Prove that the circumcircle of triangle $CDF$ contains the foot of the height of triangle $ABC$ from vertex $A$.