Generalizations of Triangle Relations to Spherical and Hyperbolic Geometry
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We generalize relations known in Euclidean geometry to spherical and hyperbolic geometry. First we prove a theorem which produces analogues for certain Euclidean inequalities involving a triangle's circumradius, inradius, and side lengths. This theorem is applied to strengthenings of Euler's inequality, R >2r. We generalize, to spherical geometry, the extension of Euler's inequality to a simplex in n-dimensional space. We prove a simple relation between the pairwise distances generated by n + 2 points in n-dimensional space of any curvature, using Cayley-Menger determinants. Finally, Euler's theorem relating the circumradius and inradius with the distance between the circumcenter and incenter is generalized to all three geometries.