On the Existence of Symmetric Bicircular Central Configurations of the 3n-Body Problem

In this paper, we consider central configurations of the planar 3n-body problem consisting n masses at vertices a regular n-gon inscribed in circle radius r and 2n second (not necessarily regular) concentric 2n-gon ar which are symmetric sense that set positions 3n corresponding invariant under action finite subgroup O(2). There two different types such configurations. first type, called bicircular problem, is regular, aligned with alternate values. semiregular its all them equal. A has pair by reflection an angle \(\beta \) respect to axis symmetry n-gon. Our aim analyze values parameter for parameters \((a,\beta )\) providing particular, \(n\ge 2 prove analytically existence (respectively )\)) satisfying some particular conditions. Using either computer-assisted results or numerical results, also describe complete \(n=2,3,4,5\) give evidences pattern \(n>5\) same as one \(n=5\).