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
Incircle and excircles of a triangle using the side lengths of the triangle relative to the perimeter—i.e., using the barycentric coordinates given above, normalized to sum to unity—as weights. (The weights are positive so the incenter lies inside the triangle as stated above.) If the three vertices are located at formula_14, formula_15, and formula_16, and the sides opposite these vertices have corresponding lengths formula_7, formula_8, and formula_9, then the incenter is at Denoting the incenter of triangle "ABC" as "I", the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation Additionally, where "R" and "r" are the