Geometry · Triangle centres · Euler line · Vectors · Homothety
Let \(ABC\) be a triangle with \(BC \neq CA\), and let \(H\), \(T\) and \(O\) be its orthocentre, centroid and circumcentre. Let \(P\) be the point symmetric to \(T\) with respect to \(O\), and let \(Q\) be the point symmetric to \(H\) with respect to \(O\). Let \(D\) be the midpoint of \(AB\), let \(R\) be the centroid of the triangle \(ABQ\), and let \(U\) be the intersection point of the lines \(OD\) and \(RT\). Prove that \(U\) is the centroid of the triangle \(DPT\).
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Serbian Regional Competition 2010, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source