Special right triangle triangles are used to aid in calculating common trigonometric functions, as below: The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the equilateral/equiangular (60°–60°–60°) triangle are the three Möbius triangles in the plane, meaning that they tessellate the plane via reflections in their sides; see Triangle group. In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or radians. Hence, the angles respectively measure 45° (), 45° (), and 90° (). The sides in this triangle are in the ratio 1 : 1
Incircle and excircles of a triangle Because the Incenter is the same distance of all sides the trilinear coordinates for the incenter are The barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions. Barycentric coordinates for the incenter are given by where formula_7, formula_8, and formula_9 are the lengths of the sides of the triangle, or equivalently (using the law of sines) by where formula_11, formula_12, and formula_13 are the angles at the three vertices. The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices