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l_1^2>\sqrt3 S>l_2^2 , triangle area inequality related to angle bisectors

Source: Sharygin 2011 Final 10.6

March 31, 2019
geometrygeometric inequalityangle bisectorarea of a triangle

Problem Statement

Prove that for any nonisosceles triangle l12>3S>l22l_1^2>\sqrt3 S>l_2^2, where l1,l2l_1, l_2 are the greatest and the smallest bisectors of the triangle and SS is its area.