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In the planar three-body problem under Newtonian potential, it is well knownthat any masses, located at the vertices of an equilateral triangle generates arelative equilibrium, known as the Lagrange relative equilibrium. In fact, theequilateral triangle is the unique mass independent shape for a relativeequilibrium in this problem.  The two dimensional positive curved three-body problem, is a naturalextension of the Newtonian three-body problem to the sphere $\mathbb{S}^2$,where the masses are moving under the influence of the cotangent potential.S.~Zhu showed that in this problem, equilateral triangle on a rotating meridiancan form a relative equilibria for any masses. This was the first report ofmass independent shape on $\mathbb{S}^2$ which can form a relative equilibrium.%  In this paper, we show that, in addition to the equilateral triangle, threeisosceles triangles on a rotating meridian, with$\theta_{ij}=\theta_{jk}=2^{-1}\arccos((\sqrt{2}-1)/2)$ for $(i,j,k)=(1,2,3)$,$(2,3,1)$, $(3,1,2)$ always form relative equilibria for any choice of themasses; where $\theta_{ij}$ are the angles between the two masses $i$ and $j$seen from the centre of $\mathbb{S}^2$. Additionally we prove that, theequilateral and the above three isosceles relative equilibria are unique withthis characteristic. We also prove that each relative equilibrium generated bya mass independent shape is not isolated from the other relative equilibria.

Mass independent shapes for relative equilibria in the two dimensional positive curved three body problem