  We classify all planar four-body central configurations where two pairs of
the bodies are on parallel lines. Using Cartesian coordinates, we show that the
set of four-body trapezoid central configurations with positive masses forms a
two-dimensional surface where two symmetric families, the rhombus and isosceles
trapezoid, are on its boundary. We also prove that, for a given position of the
bodies, in some cases an specific order of the masses determine the geometry of
the configuration, namely acute or obtuse trapezoid central configuration. We
also prove the existence on non-symmetric trapezoid central configuration with
two pairs of equal masses.
