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MA + MB + MC <=max (AB + BC, BC + AC, AC + AB) for interior M in ABC

Source: 2005 Oral Moscow Geometry Olympiad grades 10-11 p5

October 21, 2020
geometrygeometric inequality

Problem Statement

An arbitrary point MM is chosen inside the triangle ABCABC. Prove that MA+MB+MCmax(AB+BC,BC+AC,AC+AB)MA + MB + MC \le max (AB + BC, BC + AC, AC + AB).
(N. Sedrakyan)