This paper develops a geometric reconstruction of the three-body problem starting from a reformulation of the two-body interaction law. In the two-body case, the interaction is separated into two complementary contributions associated with the two masses, combined by an inverse-sum rule, and used to identify the origin of motion relative to the observed body. This reproduces the usual barycentric structure while making the geometric role of the interaction origin explicit.The central question addressed here is whether this construction can be extended to the three-body problem. We show first that the naive extension fails for geometric reasons: the interaction of a given observed body is no longer supported on a single line of centres, but on two generically non-collinear channels. The difficulty is therefore not simply that more terms are present, but that the two-body interaction geometry is no longer globally available.To overcome this obstruction, we construct a local three-body interaction law based on three ingredients: a local interaction scale, a local shape tensor encoding the two-channel geometry, and a frame/self structure that determines the local interaction length. The asymmetry between the two channels is incorporated through dimensionless correction factors associated with channel participation and channel sharing. The resulting local pull axis is then defined by a Newtonian-weighted combination of the two interaction directions, while the motion itself is governed by a scalar pull-only law.The construction is tested on the principal benchmark solutions. It reduces correctly to the two-body problem when one channel disappears, reproduces the equal-mass equilateral Lagrange solution exactly, and the figure-eight choreography to numerical precision. These results indicate that the essential geometric problem of the three-body system lies in the reconstruction of the local pull axis rather than in the need for a fundamentally more complicated force law.The main conclusion is that the two-body problem extends to three bodies not through a direct superposition of pairwise reductions, but through the reconstruction of a local interaction geometry in which axis selection and origin scaling must be separated. Within this framework, a scalar pull-only law becomes sufficient once the correct local pull axis and origin has been identified.