A triangle with side lengths , , and and triangle area satisfiesEquality holds iff the triangle is equilateral.
Ono (1914) conjectured that the inequalityholds true for all triangles, where , , and are the lengths of the sides and is the area of the triangle. This conjecture was shown to be false by Quijano (1915), although it was subsequently proved to be true for acute triangles by Balitrand (1916). A simple counterexample is provided by the triangle with , , and .
The Euler triangle formula states that the distance between the incenter and circumcenter of a triangle is given bywhere is the circumradius and is the inradius. This immediately gives the inequalitywhere equality holds iff the triangle is an equilateraltriangle.This inequality was published by Euler in 1765 (Bottema et al. 1969, p. 48).Mitrinovic et al. (1989) refer to it as the Chapple-Euler inequality.
Let be the volume of a ball of radius in a complete -dimensional Riemannian manifold with Ricci curvature tensor . Then , where is the volume of a ball in a space having constant sectional curvature. In addition, if equality holds for some ball, then this ball is isometric to the ball of radius in the space of constant sectional curvature .
Any bounded planar region with positive area placed in any position of the unit square lattice can be translated so that the number of lattice points inside the region will be at least (Blichfeldt 1914, Steinhaus 1999). The theorem can be generalized to dimensions.