On The Geometry of a Triangle in the Elliptic and in the Extended Hyperbolic Plane

TL;DR

This paper explores various classical and modern triangle centers and configurations within elliptic and extended hyperbolic geometries using a unified metric framework.

Contribution

It provides a comprehensive analysis of triangle centers and related geometric objects in elliptic and hyperbolic planes, extending classical Euclidean concepts.

Findings

Identification of centers based on orthogonality and their properties

Analysis of radical centers and centers of similitude in non-Euclidean geometries

Description of substitutes for the Euler line in elliptic and hyperbolic contexts

Abstract

We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane, such as: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude, orthology, Kiepert perspectors and related objects, Tucker circles, isoptics, substitutes for the Euler line. For both, the elliptic and the extended hyperbolic plane, a uniform metric is used.