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.

References in corpus (3)

• Convex Four Body Central Configurations with Some Equal Masses
• A Four-Body Convex Central Configuration with Perpendicular Diagonals Is Necessarily a Kite
• Four-body Central Configurations with Adjacent Equal Masses

Cited by in corpus (1)

• Four-body central configurations with one pair of opposite sides parallel