Problem

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2017 Shortlist G4 day2

Tags: geometry, arc midpoint, circumcircle



In triangle $ABC$ with incenter $I$ and circumcircle $\omega$, the tangent through $C$ to $\omega$ intersects $AB$ at point $D$. The angle bisector of $\angle CDB$ intersects $AI$ and $BI$ at $E$ and $F$, respectively. Let $M$ be the midpoint of $[EF]$. Prove that line $MI$ passes through the midpoint of arc $ACB$ of $w$ .