Let ABC be an acute-angled triangle. Let L be any line in the plane of the triangle ABC. Denote by u, v, w the lengths of the perpendiculars to L from A, B, 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 is the area of the triangle ABC. Determine the lines L for which equality holds. trigonometrygeometryfunctiongeometric inequalityarea of a triangleIMO Shortlist