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IMO Shortlist 2008, Geometry problem 6

Source: IMO Shortlist 2008, Geometry problem 6, German TST 7, P3, 2009, Exam set by Christian Reiher

July 9, 2009

geometrycircumcirclesymmetryhomothetyquadrilateralIMO Shortlist

Problem Statement

There is given a convex quadrilateral

ABCD

. Prove that there exists a point

P

inside the quadrilateral such that \angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ} if and only if the diagonals

AC

and

BD

are perpendicular.
Proposed by Dusan Djukic, Serbia

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