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No triangle with heights as sides

Source: VI Caucasus Mathematical Olympiad

March 14, 2021
geometryGeometry inequality

Problem Statement

A triangle Δ\Delta with sidelengths a≤b≤ca\leq b\leq c is given. It appears that it is impossible to construct a triangle from three segments whose lengths are equal to the altitudes of Δ\Delta. Prove that b2>acb^2>ac.
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