In the planar three-body problem under Newtonian potential, it is well known that any masses, located at the vertices of an equilateral triangle generate a relative equilibrium, known as the Lagrange relative equilibrium. In fact, the equilateral triangle is the unique mass-independent shape for a relative equilibrium in this problem. The two-dimensional positively curved three-body problem is a natural extension of the Newtonian three-body problem to the sphere \({\mathbb {S}}^2\) , where the masses are moving under the influence of the cotangent potential. Zhu showed that in this problem, an equilateral triangle on a rotating meridian can form a relative equilibria for any masses. This was the first report of a mass-independent shape on \({\mathbb {S}}^2\) which can form a relative equilibrium. In this paper, we show that, in addition to the equilateral triangle, there exists one isosceles triangle on a rotating meridian, with two equal angles seen from the center of \({\mathbb {S}}^2\) given by \(2^{-1}\arccos ((\sqrt{2}-1)/2)\) , which always form a relative equilibrium for any choice of the masses. With this shape, there are three different mass distributions, one for each mass placed at the vertex of the triangle with a different angle. Additionally we prove that the equilateral and the above isosceles relative equilibrium are unique with this characteristic. We also prove that each relative equilibrium generated by a mass-independent shape is not isolated from the other relative equilibria.
