A homographic solution is one in which the configuration of all the masses
is preserved for all time. One example of this are the relative equilibrium
solutions. Each mass rotates at a fixed distance from the center of mass at
constant angular velocity. For the 3BP these are the colinear solutions
found by Euler and the equilateral triangle configurations found by
Lagrange.
A homothetic solution is one in which the scale size changes while retaining
the configuration and no rotation is permitted. An example of this would be
a total symmetric collapse: start 3 equal masses from rest at the vertices of
an equilateral triangle and they will retain the configuration as they
collapse to the origin.
Both of these solution types are examples of central configurations. Saari
talks about these things extensively in his book: "Collisions, Rings, and other
Newtonian N-Body Problems"