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P lies in diagonal CE of ABCDE iff (BCD)+(ADE)=(ABD)+(ABP)

Source: IMO 2019 SL G5

September 22, 2020
geometryIMO ShortlistIMO Shortlist 2019

Problem Statement

Let ABCDEABCDE be a convex pentagon with CD=DECD= DE and ∠EDC≠2⋅∠ADB\angle EDC \ne 2 \cdot \angle ADB. Suppose that a point PP is located in the interior of the pentagon such that AP=AEAP =AE and BP=BCBP= BC. Prove that PP lies on the diagonal CECE if and only if area (BCD)(BCD) + area (ADE)(ADE) = area (ABD)(ABD) + area (ABP)(ABP).
(Hungary)
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