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Median Inequalities

Source: 2011 IrMO Paper 1 Problem 2

February 8, 2018
geometryinequalities

Problem Statement

Let ABCABC be a triangle whose side lengths are, as usual, denoted by a=∣BC∣,a=|BC|, b=∣CA∣,b=|CA|, c=∣AB∣.c=|AB|. Denote by ma,mb,mcm_a,m_b,m_c, respectively, the lengths of the medians which connect A,B,CA,B,C, respectively, with the centers of the corresponding opposite sides.
(a) Prove that 2ma<b+c2m_a<b+c. Deduce that ma+mb+mc<a+b+cm_a+m_b+m_c<a+b+c. (b) Give an example of (i) a triangle in which ma>bcm_a>\sqrt{bc}; (ii) a triangle in which ma≤bcm_a\le \sqrt{bc}.
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