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Geometric inequality involving lenghts of bisectors

Source: Moldova IMO-BMO TST 2003, day 2, problem 2.

August 15, 2008
inequalitiesgeometrycircumcircleinequalities proposed

Problem Statement

Consider the triangle ABC ABC with side-lenghts equal to a,b,c a,b,c. Let p\equal{}\frac{a\plus{}b\plus{}c}{2}, R R-the radius of circumcircle of the triangle ABC ABC, r r-the radius of the incircle of the triangle ABC ABC and let la,lb,lc l_a,l_b,l_c be the lenghts of bisectors drawn from A,B A,B and C C, respectively, in the triangle ABC ABC. Prove that: l_al_b\plus{}l_bl_c\plus{}l_cl_a\leq p\sqrt{3r^2\plus{}12Rr} Proposer: Baltag Valeriu
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