In the paper the authors affirmatively answer the conjecture that all nondegenerate relative equilibria in the planar \(n\)-body problem can be continued to surfaces of constant nonzero curvature for all small positive and negative values of the curvature. Two proofs of the conjecture are given, one using Palais slicing coordinates and the other Lagrange multipliers. A numerical continuation procedure is presented and applied to Lagrange’s triangle configuration with equal masses \(m_1=m_2=m_3 = 1\) and with unequal masses \(m_1=1\), \(m_2=2\), and \(m_3=3\), and to the kite configuration with masses \(m_1=1/2\) and \(m_2=m_3=m_4=1\) arranged as a convex parallelogram, as a concave parallelogram with \(m_1\) outside the triangle form by the three equal masses, and as a concave parallelogram with \(m_1\) inside the triangle form by the three equal masses. In these five numerical continuations the interval of curvature values for which numerical continuation occurred are \([-200,0.48]\), \([-200,0.19]\), \([-200,0.24]\), \([-24.48,0.22]\), and \([-0.56,0.29]\), where the lower bound of \(-200\) was a value where the numerical continuation procedure was terminated and so it doesn’t correspond to a possible bifurcation. The paper concludes with a comparison of the Lagrange triangle solutions and the kite solutions on the same common surface of a given constant curvature, either a sphere or a hyperboloid. This view of things shows that the three bodies of the Lagrange triangle solution of unequal mass move on circles with different heights, which differs from the equal mass Lagrange triangle solution in which all the masses move on the same circle.