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inequality between sides of a quadrilateral

Source: IMO Shortlist 2006, Geometry 8

June 28, 2007
inequalitiesgeometrycircumcircletriangle inequalityIMO Shortlist

Problem Statement

Let ABCDABCD be a convex quadrilateral. A circle passing through the points AA and DD and a circle passing through the points BB and CC are externally tangent at a point PP inside the quadrilateral. Suppose that ∠PAB+∠PDC≤90∘and∠PBA+∠PCD≤90∘.\angle{PAB}+\angle{PDC}\leq 90^\circ\qquad\text{and}\qquad\angle{PBA}+\angle{PCD}\leq 90^\circ. Prove that AB+CD≥BC+ADAB+CD \geq BC+AD.
Proposed by Waldemar Pompe, Poland
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