If $x, y, z$ are the sides of a triangle, then prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$
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2026-04-02 09:34:37.1775122477
Prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$
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The given inequality may be written in the form $|(x − y)(y − z)(z − x)| < xyz.$ Since $x, y, z$ are the sides of a triangle, we know that $|x − y| < z, |y − z| < x$ and $|z − x| < y.$ Multiplying these, we obtain the required inequality.