Pedoe's inequality

The inequality of Pedoe, named after Daniel Pedoe, is a geometric statement about the side lengths and the surface areas of two triangles.

A, b and c are side lengths of a triangle with the surface area F and A, B and C, the side lengths of a triangle with the other area F, the following inequality holds:

Here the equality sign applies exactly if the two triangles are similar to each other.

Note that the calculation expression on the left side, not only with respect to the six permutations of the set { (A, A ), (B, b), (c, c ) } of pairs is symmetrical, but also - perhaps less apparent - with respect to the permutation of A with a, B with b and C c. In other words, it is a symmetric function of the given pair of triangles.

This inequality generalizes the inequality of Weitzenböck and the inequality of Hadwiger - Finsler.