  The configuration of a homothetic motion in the N-body problem is called a
central configuration. In this paper, we prove that there are exactly three
planar non-collinear central configurations for masses x, -x, y, -y with x
different from y (a parallelogram and two trapezoids) and two planar
non-collinear central configurations for masses x, -x, x, -x (two diamonds).
Except the case studied here, the only known case where the four-body central
configurations with non-vanishing masses can be listed is the case with equal
masses (Albouy, 1996), which requires the use of a symbolic computation
program. Thanks to a lemma used in the proof of our result, we also show that a
co-circular four-body central configuration has non-vanishing total mass or
vanishing multiplier.
