circumellipse, which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area. The Kiepert hyperbola is the unique conic which passes through the triangle's three vertices, its centroid, and its circumcenter. Of all triangles contained in a given convex polygon, there exists a triangle with maximal area whose vertices are all vertices of the given polygon. Specifying the location of a point in a triangle One way to identify locations of points in (or outside) a triangle is to place the triangle in an arbitrary location and orientation in the Cartesian plane, and to use Cartesian coordinates. While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane. Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which give a congruent triangle, or even by rescaling it to give a similar triangle: Trilinear coordinates specify the relative distances of a point from the sides, so that coordinates indicate that the ratio of the distance of the point from the first side to its distance from the second side is , etc. Barycentric coordinates of the form specify the point's location by the relative weights that would have to be put on the three vertices in order to balance the otherwise weightless triangle on the given point. Non-planar triangles A non-planar triangle is a triangle which is not contained in a (flat) plane. Some examples of non-planar triangles in non-Euclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry. While the measures of the internal angles in planar triangles always sum to 180°, a hyperbolic triangle has measures of angles that sum to less than 180°, and a spherical triangle has measures of angles that sum to more than 180°. A hyperbolic triangle can be obtained by drawing on a negatively curved surface, such as a saddle surface, and a spherical triangle can be obtained by drawing on a positively curved surface such as a sphere. Thus, if one draws a giant triangle on the surface of the Earth, one will find that the sum of the measures of its angles is greater than 180°; in fact