2009 IMO Problems/Problem 2
Let be a triangle with circumcentre . The points and are interior points of the sides and respectively. Let and be the midpoints of the segments and , respectively, and let be the circle passing through and . Suppose that the line is tangent to the circle . Prove that .
Author: Sergei Berlov, Russia
By parallel lines and the tangency condition, Similarly, so AA similarity implies Let denote the circumcircle of and its circumradius. As both and are inside
It follows that