“…Configurations on face IV or V, respectively, correspond to equilibria of the planar, circular, restricted four-body problem with infinitesimal mass m 3 or m 4 , respectively [7,8,20]. Configurations on face III or VI, respectively, correspond to relative equilibria of the (1 + 3)-body problem, where a central mass (body 1 or 2, respectively) is equidistant from three infinitesimal masses [9,16,26]. Note that we have not made any assumptions on the relative size of the masses.…”