MathDB

G1

Part of 2008 Balkan MO Shortlist

Problems(1)

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

4/6/2020
In acute angled triangle ABCABC we denote by a,b,ca,b,c the side lengths, by ma,mb,mcm_a,m_b,m_c the median lengths and by rbc,rca,rabr_{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 ma2rbc+mb2rab+mc2rab≥2738abc3\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}
geometric inequalitygeometrymedianexradiusexcircle