Geometry · Orthocentre · Circumcircle · Reflection of the orthocentre · Antipode · Parallelogram · Midline
Problem 2, 2011
Let \(H\) and \(O\) be the orthocentre and the circumcentre of a triangle \(ABC\) with \(AB \neq AC\). The lines \(AH\) and \(AO\) meet the circumcircle of \(ABC\) a second time at \(M\) and \(N\) respectively. Let
\[ P = BC \cap HN, \qquad Q = BC \cap OM, \qquad R = HQ \cap OP . \]
Prove that the quadrilateral \(AORH\) is a parallelogram.
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Serbian National Competition (Drzavno takmicenje) 2011, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source