Problem

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

Tags: geometric inequality, geometry



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