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Σ \sqrt{\frac{ab(p- c)}{p}} \ge 6r, geometric inequality

Source: 2009 Sharygin Geometry Olympiad Final Round problem 1 grade 10

July 26, 2018
geometric inequalitygeometry

Problem Statement

Let a,b,ca, b, c be the lengths of some triangle's sides, p,rp, r be the semiperimeter and the inradius of triangle. Prove an inequality ab(p−c)p+ca(p−b)p+bc(p−a)p≥6r\sqrt{\frac{ab(p- c)}{p}} +\sqrt{\frac{ca(p- b)}{p}} +\sqrt{\frac{bc(p-a)}{p}} \ge 6r
(D.Shvetsov)
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