Is it obvious or easily-proven that L4 and L5 must be in the parental orbital plane?

Question

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.

Answer 1

Is it obvious or easily-proven that L4 and L5 must be in the parental orbital plane?

Yes it is fairly easily proven. At the Lagrange points, forces cancel to zero in the rotating frame.

If you look at a point above the plane, there will be a force downward from both objects, and there's nothing above to cancel that, so there can not be a stationary or Lagrange point out of plane.

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?

That's a different question than the title. There are all kinds of different halo orbits that can be associated with Lagrange points. They can circle around the points in-plane, or out of plane. Some examples can be found in this answer and in this question.