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, $U={\alpha }^{-1}\sum m_i{m}_j\mathrm{/}{r}_{ij}^{\alpha }$ for $\alpha\ne 0$,amotionhavingconstantconfigurationalmeasure$\mu=I^{\alpha/2}U$ is homographic, where $I$ represents the moment of inertia defined by $I=\sum m_i m_j r_{ij}^2/\sum m_k$,$m_i$themass,and$r_{ij}$ the distance between particles. We prove this conjecture for general masses $m_k>0$ in the planar three-body problem under Newton potential ($\alpha =1$) and a strong force potential ($\alpha =2$).