Planarity conditions and four-body central configurations equations with angles as coordinates

TL;DR

This paper introduces new angle-based equations for four-body central configurations, providing geometric insights and novel proofs, including an explanation of Ptolemy's theorem's role in co-circular configurations.

Contribution

It develops new equations using angles for four-body central configurations and offers geometric explanations and proofs, enhancing understanding of these configurations.

Findings

Derived new angle-based equations for four-body configurations

Provided geometric explanations for Ptolemy's theorem in this context

Offered novel proofs of existing results in four-body central configurations

Abstract

We discuss several conditions for four points to lie on a plane, and we use them to find new equations for four-body central configurations that use angles as variables. We use these equations to give novel proofs of some results for four-body central configuration. We also give a clear geometrical explanation of why Ptolemy's theorem can be used to write equations for co-circular central configurations when mutual distances are used as coordinates.