MathNet 2019-part-05 — 184 problems (EN–VI) Question 21

Let $ABC$ be a triangle, and let $I$ and $O$ be its incenter and circumcenter, respectively. The $A$-excircle touches the lines $AB$, $AC$, $BC$ at $K$, $L$, $M$, respectively. Show that, if the midpoint of the segment $KL$ lies on the circle $ABC$, then $I$, $M$, $O$ are collinear.

Pavel Kozhevnikov, Russian Olympiad, 2005