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\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1

Source: Polish MO Second Round 1973 p1

September 8, 2024
algebrainequalitiesgeometryGeometric Inequalities

Problem Statement

Prove that if positive numbers x,y,z x, y, z satisfy the inequality x2+y2−z22xy+y2+z2−x22yz+z2+x2−y22xz>1, \frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1, then they are the lengths of the sides of a certain triangle.
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