http://www.merlyn.demon.co.uk/gravity6.htm is a somewhat strange web page by "Dr J.R. Stockton," also with contributions by Michael Behrend, plus looking into original works by Euler & Lagrange, which outlines a simple proof that with 3 point bodies of arbitrary positive masses A,B,C the only solutions of the 3 body problem in which the configuration stays unaltered in shape, is where the bodies are (i) collinear or (ii) vertices of equilateral triangle. Note, the masses not required to be equal, but you still get equilateral triangle (it rotates about its center of mass) and the force not required to be inverse-square (any of a wide class of functions of distance will do), and his page has scroll-windows within scroll-windows making it a bit strange to read. He does not include a stability analysis. It seems to me the collinear solutions are always unstable even to perturbations staying on the line. The triangle solutions might sometimes be stable. It's not obvious how to prove stability. Proving instability can sometimes be easy, since you just present some instability mechanism; but stability seems much harder, since need to disprove every possible instability. Stockton also has a few clearly incomplete mutterings about stationary-shape 4-body problem. For example, given 4 bodies of masses A,A,B,B you can get a rotating rhombus stationary-shape 4-body solution. He does not mention, but for any 4 masses A,B,C,D you can put A,B,C in the equilateral triangle solution and D at the center of mass. Another he does not mention: given 4 bodies A,B,C,C, you can place C,C at the 2 far image-corners of a bilaterally-symmetric quadrilateral and A,B at the other two corners and rotate thing about its center of mass, and this will work if rotation-speed and quad-shape are chosen right. For this one you can also place a fifth body D at the center of mass. --- EQUIVALENT PROBLEM: It actually has advantages to think about this whole problem in the "opposite" way. That is, instead of the centrifugal repulsive force, consider an attractive "spring" potential of form r*r*k for r=distance to rotation-center and k=positive constant. And instead of the gravitational inter-body attractions from positive mass-points, consider Coulomb repulsions (of positive charge-points). If you just attach your charges to a center point with a spring (or place in a shallow parabolic dish) and let them settle, they clearly must+will find a position of local-min energy. This proves EXISTENCE THEOREM: for any N>0, and any N positive real masses M1,M2,M3,...,MN, there exists at least one configuration in the plane (i.e. locations for each mass) such that that configuration is a solution of the Newton gravity+motion laws which just rotates bodily, preserving its shape & size. This is very cute piece of jujitsu, eh? In the original problem formulation, this is not obvious, but I simply negate the potential function, and voila, existence becomes obvious. The bad news is: every solution generated in this "negated" way, since a min of the negated-potential, automatically is a max of the original pseudo-potential, hence automatically is unstable in the original context? There are two reasons I am putting "?": 1. conservation of angular momentum and center of mass perhaps will enable stability even though without those conservation laws we would have instability. 2. A max of a potential is clearly unstable, but for a PSEUDO-potential, perhaps you can still sometimes survive. This "2" represents a flaw in my original "Theorem" that every N-body stationary-shape solution if N>6 is automatically unstable -- that proof has now been murdered by several such flaws (groan!) but might still be correct! (And stable solutions in the original Newton context are necessarily unstable in the Coulombized opposite version? But instability in the Coulomb setting might not imply stability in the Newton setting!) So this suggests, but in view of caveat 1 & 2 does not prove, that: The equilateral triangle 3-body solutions always are unstable.