  In the $2$-dimensional $n$-body problem, $n\ge 3$, in spaces of constant
curvature, $\kappa\ne 0$, we study polygonal homographic solutions. We first
provide necessary and sufficient conditions for the existence of these orbits
and then consider the case of regular polygons. We further use this criterion
to show that, for any $n\ge 3$, the regular $n$-gon is a polygonal homographic
orbit if and only if all masses are equal. Then we prove the existence of
relative equilibria of non-equal masses on the sphere of curvature $\kappa>0$
for $n=3$ in the case of scalene triangles. Such triangular relative equilibria
occur only along fixed geodesics and are generated from fixed points of the
sphere. Finally, through a classification of the isosceles case, we prove that
not any three masses can form a triangular relative equilibrium.
