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A complete quadrangle problem with bisectors and diagonals

Source: Brazilian Math Olympiad 2007, Problem 5

November 2, 2007
geometry unsolvedgeometry

Problem Statement

Let ABCD ABCD be a convex quadrangle, P P the intersection of lines AB AB and CD CD, Q Q the intersection of lines AD AD and BC BC and O O the intersection of diagonals AC AC and BD BD. Show that if \angle POQ\equal{} 90^\circ then PO PO is the bisector of ∠AOD \angle AOD and OQ OQ is the bisector of ∠AOB \angle AOB.
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