We present a geometric approach to the three-body problem in the non-relativistic context of the Barbour-Bertotti theories. The Riemannian metric characterizing the dynamics is analysed in detail in terms of the relative separations. Consequences of a conformal symmetry are exploited and the sectional curvatures of geometrically preferred surfaces are computed. The geodesic motions are integrated. Line configurations, which lead to curvature singularities for N 3, are investigated. None of the independent scalars formed from the metric and curvature tensor diverges there.