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x^2+y^2+z^2 >= x^3+y^3+z^3 +6xyz, inside an equilateral of height 1

Source: Mediterranean Mathematical Olympiad 2019 P4 MMC

July 21, 2019
geometric inequalitygeometryinequalitiesdistance

Problem Statement

Let PP be a point in the interior of an equilateral triangle with height 11, and let x,y,zx,y,z denote the distances from PP to the three sides of the triangle. Prove that x2+y2+z2 ≥ x3+y3+z3+6xyz x^2+y^2+z^2 ~\ge~ x^3+y^3+z^3 +6xyz
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