On Centers and Central Lines of Triangles in the Elliptic Plane

TL;DR

This paper explores triangle centers in elliptic geometry, focusing on lines analogous to Euler and Brocard lines, and investigates elliptic counterparts to classical Euclidean circles.

Contribution

It determines barycentric coordinates of triangle centers in the elliptic plane and studies elliptic analogs of classical Euclidean circles and lines.

Findings

Identified elliptic analogs of Euler and Brocard lines.

Derived barycentric coordinates for triangle centers in elliptic geometry.

Explored elliptic substitutes for Euclidean nine-point circle and other circles.

Abstract

We determine barycentric coordinates of triangle centers in the elliptic plane. The main focus is put on centers that lie on lines whose euclidean limit (triangle excess --> 0) is the Euler line or the Brocard line. We also investigate curves which can serve in elliptic geometry as substitutes for the euclidean nine-point-circle, the first Lemoine circle or the apollonian circles.