Triangle center Since rotations and translations may be regarded as double reflections they too must preserve triangle centers. These invariance properties provide justification for the definition. Some other names for dilation are uniform scaling, isotropic scaling, homothety, and homothecy. The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both Euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must "not" encapsulate the specification of the angle sum being 180 degrees.
Triangle group of these relations and that Δ("l,m,n") is a discrete group of motions of the corresponding space. Thus a triangle group is a reflection group that admits a group presentation An abstract group with this presentation is a Coxeter group with three generators. Given any natural numbers "l", "m", "n" > 1 exactly one of the classical two-dimensional geometries (Euclidean, spherical, or hyperbolic) admits a triangle with the angles (π/l, π/m, π/n), and the space is tiled by reflections of the triangle. The sum of the angles of the triangle determines the type of the geometry by the Gauss–Bonnet theorem: it