On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry

TL;DR

This paper explores hyperbolic triangles and circles using barycentric coordinates in relativistic hyperbolic geometry, introducing hyperbolic analogs of Euclidean theorems through the gyrogeometric framework.

Contribution

It develops hyperbolic counterparts of classical Euclidean theorems using gyrogeometry and barycentric coordinates, expanding the analytic hyperbolic geometry toolkit.

Findings

Hyperbolic versions of classical Euclidean theorems established

Introduction of gyroconcepts like gyrovector spaces and gyrolanguage

New geometric relations in relativistic hyperbolic geometry

Abstract

Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are {\it gyrogroups} and {\it gyrovector spaces}, and Einstein addition being both {\it gyrocommutative} and {\it gyroassociative}. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".