Constant Normalized Area in the Planar Three-Body Problem

We prove that a collision-free noncollinear planar Newtonian three-body motion with constant triangle area divided by barycentric moment of inertia is necessarily a Lagrange homographic motion. The masses are arbitrary positive numbers, and the conclusion holds on any open time interval. Consequently, the collision-free exactly periodic planar solutions with constant unsigned normalized area are precisely the elliptic Lagrange and Euler families, including their circular members.

In mass-Jacobi shape coordinates the area condition fixes a latitude. We continue necessary longitude, normal-force and energy identities to an isolated complex binary branch point. Two possible vanishing orders exhaust the cases and give incompatible leading energy coefficients. The reduction and classical families are standard; the proof uses an analytic energy-comparison method related to work on Saari's conjecture.

The result addresses the planar part of AIM workshop Problem 18, not its genuinely spatial extension or separate meridian question. This is an AI-assisted, self-audited, unrefereed preprint and makes no absolute-priority claim.