Looking at the question and answer and diagrams from Are Lagrangian points associated only with the smaller body? got me wondering.
L4 and L5 each form an equilateral triangle with the two main bodies, with L4 ahead in the orbit and L5 behind it.
But there's a whole circle of points equilaterally distant from the two main bodies, perpendicular to the orbital plane! Couldn't that be an orbit?
On some intuitive level I feel like that's just obviously dumb, but for something so obvious, I sure can't put my finger on it.