IMO Shortlist 2009 problem G6

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Dodao/la: arhiva
2. travnja 2012.
Let the sides AD and BC of the quadrilateral ABCD (such that AB is not parallel to CD) intersect at point P. Points O_1 and O_2 are circumcenters and points H_1 and H_2 are orthocenters of triangles ABP and CDP, respectively. Denote the midpoints of segments O_1H_1 and O_2H_2 by E_1 and E_2, respectively. Prove that the perpendicular from E_1 on CD, the perpendicular from E_2 on AB and the lines H_1H_2 are concurrent.

Proposed by Ukraine
Izvor: Međunarodna matematička olimpijada, shortlist 2009