Saari's homographic conjecture for general masses in planar three-body problem under Newton potential and a strong force potential

Saari?s homographic conjecture claims that in the N-body problem under the homogeneous potential, for , a motion with the constant configurational measure is homographic, where I represents the moment of inertia defined by , mi is the mass, and rij is the distance between particles. We prove this conjecture for general masses, in the planar three-body problem under the Newton potential () and a strong force potential ().