The central configurations of four masses x,-x,y,-y

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 not equal 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 (A. Albouy, 1995-1996), 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 nonvanishing total mass or vanishing multiplier. (c) 2007 Elsevier Inc. All rights reserved.