I wrote:

What I ideally want is a set of three parameters such that the distances between all the points can be expressed as rational functions of the parameters and the original lengths AB, AC, BC. Is this too much to hope for?

I should say that if I can't have this, I'd settle for a k-parameter sub-family of the three-parameter family with the property that all distances (for configurations within the sub-family) can be expressed as rational functions of the k parameters and the lengths AB, AC, BC. I'd even be okay with k=0. That is, I'd settle for knowing ONE way of choosing P,D,E,F so that all the distances are rational functions of the lengths AB, AC, BC. (But this seems even less likely to me.) As you might guess from the odd specificity of my question, there's some context for this problem. The answer might be very useful to me as a source of analogy for some research I'm doing in algebraic combinatorics, or it might turn out to be a dead-end, but in any case the context isn't very math-fun-ish, so I'll spare you the details (except to say that it has to do with things called "cluster algebras" that have been turning up lately in a bunch of different fields of mathematics). Jim Propp