Pedoe's inequality

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In geometry, Pedoe's inequality, named after Daniel Pedoe, states that if a, b, and c are the lengths of the sides of a triangle with area ƒ, and A, B, and C are the lengths of the sides of a triangle with area F, then

A^2(b^2+c^2-a^2)+B^2(a^2+c^2-b^2)+C^2(a^2+b^2-c^2)\geq 16Ff,\,

with equality if and only if the two triangles are similar.

The expression on the left is not only symmetric under any of the six permutations of the set { (Aa), (Bb), (Cc) } of pairs, but also—perhaps not so obviously—remains the same if a is interchanged with A and b with B and c with C. In other words, it is a symmetric function of the pair of triangles.

Pedoe's inequality is a generalization of Weitzenböck's inequality and of the Hadwiger–Finsler inequality.

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