KöMaL 2025 B5489

In right triangle $ABC$ the angles are the following: $\angle ABC=15^\circ$ and $\angle CAB=75^{\circ}$. Let $F$ be the midpoint of hypotenuse $AB$. Let point $D$ be chosen on leg $BC$ such that $BD=CA$, and let point $E$ be chosen on ray $CA$ beyond point $A$ such that $CE=BC$. Let $M$ be the point of intersection of lines $BE$ and $CF$. Prove that lines $DM$ and $CM$ are tangent to the circumcircle of triangle $AEF$.