Problem

Source: ELMO Shortlist 2013: Problem G7, by Michael Kural

Tags: ratio, geometry, parallelogram, circumcircle, trigonometry, collinear, complex



Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. Proposed by Michael Kural