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Nice and interesting JBMO problem

Source: JBMO Shortlist 2014,G6

November 6, 2016
geometry

Problem Statement

Let ABCDABCD be a quadrilateral whose diagonals are not perpendicular and whose sides ABAB and CDCD are not parallel.Let OO be the intersection of its diagonals.Denote with H1H_1 and H2H_2 the orthocenters of triangles AOBAOB and COD,COD, respectively.If MM and NN are the midpoints of the segment lines ABAB and CD,CD, respectively.Prove that the lines H1H2H_1H_2 and MNMN are parallel if and only if AC=BD.AC=BD.
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