Chromogeometry

TL;DR

Chromogeometry explores the interplay of Euclidean and relativistic geometries through a three-fold symmetric framework, revealing new interactions among classical triangle centers using rational trigonometry.

Contribution

It introduces a novel framework combining Euclidean and relativistic geometries with rational trigonometry, uncovering new relationships among triangle centers.

Findings

Red and green Euler lines interact with blue ones.

The three orthocenters form a triangle with notable collinearities.

Relativistic geometries are effectively analyzed using quadrance and spread.

Abstract

Chromogeometry brings together Euclidean geometry (called blue) and two relativistic geometries (called red and green), in a surprising three-fold symmetry. We show how the red and green `Euler lines' and `nine-point circles' of a triangle interact with the usual blue ones, and how the three orthocenters form an associated triangle with interesting collinearities. This is developed in the framework of rational trigonometry using quadrance and spread instead of distance and angle. The former are more suitable for relativistic geometries.