Planar Central Configurations of Symmetric Five-Body Problems with Two Pairs of Equal Masses

We study central configuration of a set of symmetric planar five-body problems where (1) the five masses are arranged in such a way that m(1), m(2), and m(4) are collinear and m(2), m(3), and m(5) are collinear; the two sets of collinear masses form a triangle with m(2) at the intersection of the two sets of collinear masses; (2) four of the bodies are on the vertices of an isosceles trapezoid and the fifth body can take various positions on the axis of symmetry both outside and inside the trapezoid. We form expressions for mass ratios and identify regions in the phase space where it is possible to choose positive masses which will make the configuration central. We also show that the triangular configuration is not possible.