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Equal distances between pairs of orthocenters in cyclic quad

Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level

July 31, 2011
geometrycircumcirclegeometric transformationreflectioncyclic quadrilateralgeometry unsolved

Problem Statement

In a cyclic quadrilateral ABCDABCD, the extensions of sides ABAB and CDCD meet at point PP, and the extensions of sides ADAD and BCBC meet at point QQ. Prove that the distance between the orthocenters of triangles APDAPD and AQBAQB is equal to the distance between the orthocenters of triangles CQDCQD and BPCBPC.