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We study four-body central configurations with one pair of opposite sidesparallel. We use a novel constraint to write the central configurationequations in this special case, using distances as variables. We prove that,for a given ordering of the mutual distances, a trapezoidal centralconfiguration must have a certain partial ordering of the masses. We also showthat if opposite masses of a four-body trapezoidal central configuration areequal, then the configuration has a line of symmetry and it must be a kite. Incontrast to the general four-body case, we show that if the two adjacent massesbounding the shortest side are equal, then the configuration must be anisosceles trapezoid, and the remaining two masses must also be equal.

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