KöMaL 2026 C1896

Let $E$ be the center of square $ABCD$. A quarter circle with center $D$ and radius $DA$ inside the square intersects the semicircle with diameter $AB$ inside the square at point $F$. Line $EF$ intersects line $AB$ at $M$, and the semicircle with diameter $DA$ outside the circle at point $N$. Prove that $F$ is the midpoint of line segment $MN$.