BxMO 2025 Problem 3

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$ . Let $D, E, F$ be the midpoints of the arcs $\overline{BC}, \overline{CA}, \overline{AB}$ of $\Omega$ not containing $A, B, C$ , respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$ . Show that $I, D'$ , and the midpoint $M$ of $[EF]$ lie on a line.