In this article we study the planar 4-body problem under homogeneous power-law potentials where the interaction between the bodies is given by r −a , a 4/3 (the Newtonian case corresponding to a = 3 and the vortex problem corresponding to a = 2). We study convex central configurations assuming two pairs of positive equal masses located at two adjacent vertices of a convex quadrilateral. Under these assumptions we prove that the isosceles trapezoid is the unique central configuration for every a 4/3. For the case 0 < a < 4/3, which is more difficult to study analytically, we present some numerical considerations.