Saari's homographic conjecture, which extends a classical statement proposed
by Donald Saari in 1970, claims that solutions of the Newtonian n-body
problem with constant configurational measure are homographic. In other words,
if the mutual distances satisfy a certain relationship, the configuration of
the particle system may change size and position but not shape. We prove this
conjecture for large sets of initial conditions in three-body problems given by
homogeneous potentials, including the Newtonian one. Some of our results are
true for n≥3
