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Ienquality involving sides a,b,c and angles α, β, γ

Source: Baltic Way 1994

December 22, 2011
geometry unsolvedgeometryInequality

Problem Statement

Let α,β,γ\alpha,\beta,\gamma be the angles of a triangle opposite to its sides with lengths a,b,ca,b,c respectively. Prove the inequality a(1β+1γ)+b(1γ+1α)+c(1α+1β)≥2(aα+bβ+cγ)a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)
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