Rhomboidal Central Configurations of the Five-Body Problem

In this paper we investigate rhomboidal central configurations for the planar fivebody problem. These are configurations whose convex hulls are rhombuses, and one interior mass lies on a diagonal of the rhombus. Consider partition of the configuration space by orderings of mutual distances, our main theorem provides criteria for central configurations in every region of this partition. For some regions we obtain existence and uniqueness of central configurations, for some we obtain nonexistence, for others our criteria are necessary conditions on masses. Our proofs are elementary and completely analytic.