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

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.