Duality (order theory)

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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

A2(b2+c2a2)+B2(a2+c2b2)+C2(a2+b2c2)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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