Geometry · Vectors · Centroid · Orthocenter · Euler line
Let \(O\) and \(H\) be the circumcenter and the orthocenter of a triangle \(ABC\), and let \(G_1\), \(G_2\), \(G_3\) be the centroids of the triangles \(HBC\), \(HCA\), \(HAB\), respectively. Prove that
\[ \overrightarrow{AG_1} + \overrightarrow{BG_2} + \overrightarrow{CG_3} = \frac{2}{3}\,\overrightarrow{OH}. \]
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Serbian Municipal Competition 2011, high school grade I, category A, problem 1. Organized by the Mathematical Society of Serbia (DMS). Source