I see you've found the answer to your problem. Here's my answer, if you find it helpful:

Sounds like a circle to me. You're looking for the set of all points equidistant from two points in 3D space, right? [EDIT]: See Plato's post below for a clarification of this phraseology. That's going to be a circle. If you have two points, and [MATH\mathbf{r}_{2},[/tex] with the given distance being

then let with length

By the Pythagorean theorem, the circle in question is going to be the intersection of the sphere centered at of radius , with the plane

That is, the circle is the set of all points such that

and