  In the planar three-body problem under Newtonian potential, it is well known
that any masses, located at the vertices of an equilateral triangle generates 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 positive 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.
S.~Zhu showed that in this problem, equilateral triangle on a rotating meridian
can form a relative equilibria for any masses. This was the first report of
mass independent shape on $\mathbb{S}^2$ which can form a relative equilibrium.
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  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 centre 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. 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.
