We prove for a large class of $n$-body problems, including the classical $n$-body problem, that if for a configuration of point masses the (polar) moment of inertia is constant, the configuration of the point masses constitutes that of a rotating rigid body, which thus by extension proves Saari's conjecture. Additionally, the proof given is surprisingly short and only uses theory typically provided in an introductory course on complex analysis. Finally, we remark how the main result can be generalised well beyond the confines of the structure of an $n$-body problem.
