Spatial equilateral chain central configurations of the five-body problem with a homogeneous potential

We consider central configurations of the strictly spatial five-body problem with a homogeneous potential which are equilateral chains, i.e., configurations with four sequential equilateral edges containing all five vertices. First, we prove that any such configuration must be a triangular bipyramid with an equilateral triangle base. Furthermore, we show that the masses located at the vertices of the triangle must be equal and the masses of the other two particles which are off the base also must be equal. We also found that a particular triangular bipyramid configuration with fixed masses is a central configuration for a range of homogenous potentials generalizing the Newtonian potential. Finally, we conclude that there is a unique triangular bipyramid central configuration with equal masses for these same homogenous potentials.