# 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

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 { (*A*, *a*), (*B*, *b*), (*C*, *c*) } 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.

## References

- "A Two-Triangle Inequality", Daniel Pedoe,
*The American Mathematical Monthly*, volume 70, number 9, page 1012, November, 1963. - "An Inequality for Two Triangles", D. Pedoe,
*Proceedings of the Cambridge Philosophical Society*, volume 38, part 4, page 397, 1943.