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

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