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# Inequality

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Prove that if a, b, and c are positive real numbers, then

sqrt(a^2 + ab + b^2) + sqrt(a^2 + ac + c^2) + sqrt(b^2 + bc + c^2) >= sqrt(3) (sqrt(ab) + sqrt(ac) + sqrt(bc))

When does equality occur?

Mar 10, 2023

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Here's a hint: what if you tried to glue two equilateral triangles together?

The expressions inside the radicals remind us of the Law of Cosines. For example, $\sqrt{a^2 - ab +b^2}$ can be interpreted as the length of the third side of a triangle in which two sides have lengths $a$ and $b$, and the angle between these two sides is $60^\circ$.

From these two glues triangles, what can we say about when equality occurs? When does A+B=C in a "triangle"?

Mar 10, 2023