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Geometric inequality 2

Source: IMO ShortList 1988, Problem 27, United Kingdom 4, Problem 79 of ILL

January 12, 2005
trigonometrygeometryfunctiongeometric inequalityarea of a triangleIMO Shortlist

Problem Statement

Let ABC ABC be an acute-angled triangle. Let L L be any line in the plane of the triangle ABC ABC. Denote by u u, v v, w w the lengths of the perpendiculars to L L from A A, B B, C C respectively. Prove the inequality u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S, where S S is the area of the triangle ABC ABC. Determine the lines L L for which equality holds.
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