Problem

Source: iran2004 (geo exam)

Tags: geometry, circumcircle, geometric transformation, projective geometry, trigonometry, analytic geometry, calculus



Let $ABC$ be a triangle, and $O$ the center of its circumcircle. Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively. Prove that $\measuredangle ROS=\measuredangle BAC$.