Saari's homographic conjecture for planar equal-mass three-body problem in Newton gravity

Saari's homographic conjecture in N-body problem under the Newton gravity is the following; configurational measure \mu=\sqrt{I}U, which is the product of square root of the moment of inertia I=(\sum m_k)^{-1}\sum m_i m_j r_{ij}^2 and the potential function U=\sum m_i m_j/r_{ij}, is constant if and only if the motion is homographic. Where m_k represents mass of body k and r_{ij} represents distance between bodies i and j. We prove this conjecture for planar equal-mass three-body problem. In this work, we use three sets of shape variables. In the first step, we use \zeta=3q_3/(2(q_2-q_1)) where q_k \in \mathbb{C} represents position of body k. Using r_1=r_{23}/r_{12} and r_2=r_{31}/r_{12} in intermediate step, we finally use \mu itself and \rho=I^{3/2}/(r_{12}r_{23}r_{31}). The shape variables \mu and \rho make our proof simple