MathDB

m_a^2/r_{bc}+m_b^2/r_{ab}+m_c^2/r_{ab} >= 27\sqrt3 /8 \sqrt[3]{abc}

Source: Balkan MO Shortlist 2008 G1

April 6, 2020

geometric inequalitygeometrymedianexradiusexcircle

Problem Statement

In acute angled triangle

ABC

we denote by

a,b,c

the side lengths, by

m_a,m_b,m_c

the median lengths and by

r_{b}c,r_{ca},r_{ab}

the radii of the circles tangents to two sides and to circumscribed circle of the triangle, respectively. Prove that

\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}