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
Extouch triangle Extouch triangle In geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle. The vertices of the extouch triangle are given in trilinear coordinates by: or equivalently, where "a,b,c" are the lengths of the sides opposite angles "A, B, C" respectively, The triangle's splitters are lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle; they bisect the triangle's perimeter and meet at the Nagel point. This is shown in blue and labelled "N" in the diagram. The Mandart inellipse is tangent to