CHAPTER 6 - Elementary Theorems of Hyperbolic Geometry

Abstract

Hyperbolic geometry is a particularly important non-Euclidean geometry. It shares with Euclidean geometry all of absolute geometry, that is, the part of Euclidean geometry based on the axioms in groups I through IV. Many well-known theorems of elementary geometry belong to absolute geometry. The Poincare model shows that the concept of hyperbolic geometry is meaningful. This chapter presents proofs for a number of elementary theorems of hyperbolic geometry. In hyperbolic geometry, the sum of the angles in a triangle is less than two right angles. Moreover, in hyperbolic geometry two triangles with pairwise congruent angles are congruent. In the Poincaré model, the locus of points equidistant from a point Mh is a Euclidean circle. However, the Euclidean center Mc and the hyperbolic center Mh do not coincide.