defined in terms of barycentrics instead of trilinears. In practice it isn't difficult to switch from one coordinate system to the other. Binary systems There are other center pairs besides the Fermat point and the 1st isogonic center. Another system is formed by X3 and the incenter of the tangential triangle. Consider the triangle center function given by: For the corresponding triangle center there are four distinct possibilities: cos(A) : cos(B) : cos(C) if the reference triangle is acute (this is also the circumcenter). [cos(A) + sec(B)sec(C)] : [cos(B) − sec(B)] : [cos(C) − sec(C)] if the angle at A is obtuse. [cos(A) − sec(A)] : [cos(B) + sec(C)sec(A)] : [cos(C) − sec(C)] if the angle at B is obtuse. [cos(A) − sec(A)] : [cos(B) − sec(B)] : [cos(C) + sec(A)sec(B)] if the angle at C is obtuse. Routine calculation shows that in every case these trilinears represent the incenter of the tangential triangle. So this point is a triangle center that is a close companion of the circumcenter. Bisymmetry and invariance Reflecting a triangle reverses the order of its sides. In the image the coordinates refer to the (c,b,a) triangle and (using "|" as the separator) the reflection of an arbitrary point α : β : γ is γ | β | α. If f is a triangle center function the reflection of its triangle center is f(c,a,b) | f(b,c,a) | f(a,b,c) which, by bisymmetry, is the same as f(c,b,a) | f(b,a,c) | f(a,c,b). As this is also the triangle center corresponding to f relative to the (c,b,a) triangle, bisymmetry ensures that all triangle centers are invariant under reflection. Since rotations and translations may be regarded as double reflections they too must preserve triangle centers. These invariance properties provide justification for the definition. Alternative terminology Some other names for dilation are uniform scaling, isotropic scaling, homothety, and homothecy. Non-Euclidean and other geometries The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in non-Euclidean geometry. Spherical triangle centers can be defined using spherical trigonometry. Triangle centers that have the same form for both Euclidean and hyperbolic geometry can be expressed using gyrotrigonometry. In non-Euclidean geometry, the assumption that the interior angles of the triangle sum to 180 degrees must be discarded. Centers of tetrahedra or higher-dimensional simplices can also be defined, by analogy with 2-dimensional triangles. Some centers can be extended