Four-body central configurations with some equal masses

We prove firstly that any convex non-collinear central configuration of the planar 4-body problem with equal opposite masses beta > alpha > 0, such that the diagonal corresponding to the mass a is not shorter than that corresponding to the mass, must possess a symmetry and therefore must be a kite. Then by a recent result of Bernat, Llibre and Perez-Chavela, this kite is actually a rhombus. Secondly we prove that a convex non-collinear planar 4-body central configuration with three equal masses must be a kite too. We also prove that the concave central configuration with three equal masses forming a triangle and the fourth one with any given mass in the interior must be either an equilateral triangle with the fourth mass at its geometric center, or an isosceles triangle with the fourth mass on the symmetry axis.