Question 80
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Take a point $M$ on the minor $\operatorname{arc} BC$ of $(O)$ (not coinciding with $B, C$) and denote by $D, E, F$ the reflections of $M$ in $OA, OB, OC$. Suppose that $BF$ meets $CE$ at $K$ and let $I$ be the incenter of triangle $DEF$.
1. Prove that the perpendicular bisectors of $EF$ and $IK$ meet on the circle $(O)$.
2. Prove that the three points $H, K, I$ are collinear.
1. Prove that the perpendicular bisectors of $EF$ and $IK$ meet on the circle $(O)$.
2. Prove that the three points $H, K, I$ are collinear.