Problem

Source: China Northern Mathematical Olympiad 2017

Tags: geometry, Spiral Similarity, circumcircle



Triangle \(ABC\) has \(AB > AC\) and \(\angle A = 60^\circ \). Let \(M\) be the midpoint of \(BC\), \(N\) be the point on segment \(AB\) such that \(\angle BNM = 30^\circ\). Let \(D,E\) be points on \(AB, AC\) respectively. Let \(F, G, H\) be the midpoints of \(BE, CD, DE\) respectively. Let \(O\) be the circumcenter of triangle \(FGH\). Prove that \(O\) lies on line \(MN\).