Do Lagrangian points actually maintain a fixed distance?

I was reading on up Lagrangian points and the restricted three-body problem.

From what I was able to tell, the Lagrangian points are 5 points in a two-body system such that a third body would be relatively at rest. The first three are unstable, and the last two are stable.

How is this possible? Because we know from Kepler's laws that the orbits are in the shape of ellipses with the sun at a focus, and we also know the planes sweep out equal areas in equal time and so the speed varies. And so, how can the third body have a constant distance when there is an ever-changing speed gap between the orbital speeds.

And could anyone provide (visually) an explanation for the Lagrangian points, how to deduce them, and what assumptions were made in order to deduce them?

LE: So, in the three body problem, the orbits more closely resemble circles than ellipses? And if so, is the speed relatively constant so that the difference in distance between the third and second body is negligible?

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As mentioned in that same article, L4 and L5 are only stable if the ratio of the masses of the objects exceeds $(25+3\sqrt{69})/2$, which happens to be the case for both the Sun-Earth system and the Earth-Moon system. – Chris White Aug 14 '12 at 3:27