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 the North Pole is a line of longitude; so there are right angles at the two points on the equator. Moreover, the angle at the North Pole is also 90° because the other two vertices differ by 90° of longitude. So the sum of the angles in this triangle is . The triangle encloses 1/4 of the northern hemisphere (90°/360° as viewed from the North Pole) and therefore 1/8 of the Earth's surface, so in the formula ; thus the formula correctly gives the sum of the triangle's angles as 270°. From the above angle sum formula we can also