On Geodesic Triangles in Non-Euclidean Geometry

Abstract

In this paper, we study centroids, orthocenters, circumcenters, and incenters of geodesic triangles in non-Euclidean geometry, and we discuss the existence of the Euler line in this context. Moreover, we give simple proofs of the existence of a totally geodesic 2-dimensional submanifold containing a given geodesic triangle in the hyperbolic or spherical 3-dimensional geometry.

Keywords

• non-Euclidean geometry
• notable points of hyperbolic and spherical triangles
• polar triangle
• totally geodesic hypersurfaces