Integer triangle we today use the term Heronian triangle; Somos uses it to refer to triangles whose ratios of sides are rational; Conway and Guy define a rational triangle as one with rational sides and rational angles measured in degrees—in which case the only rational triangle is the rational-sided equilateral triangle. There are various general properties for an integer triangle, given in the first section below. All other sections refer to classes of integer triangles with specific properties. Any triple of positive integers can serve as the side lengths of an integer triangle as long as it satisfies the triangle inequality: the
Elliptic geometry geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees. For sufficiently small triangles, the excess over 180 degrees can be made arbitrarily small. The Pythagorean theorem fails in elliptic geometry. In the 90°–90°–90° triangle described above, all three sides have the same length, and consequently do not satisfy formula_1. The Pythagorean result