Find the distance between the two apexes of two three-sided pyramids

[Herbascious J]

TL;DR Summary

What would be a simple (as possible) equation to determine the distance between two apexes of two three-sided pyramids which have identical bases, but unique apexes. You are only given the distances between points, no angles.

A way to imagine this problem to stand on a plane. You have chosen three points on that plane to create a triangle-base and from these three points you can make distance measurements in any direction. You know the distance between each of the three points of the base, but you do not yet know any angles, and you will not be allowed to calculate these. Above you are two points suspended high in the air. You may imagine that each of these points could be their own apex of a three sided pyramid with the triangle base you have on the ground plane. You would like to know the distance between these two apexes precisely, but you can not measure it directly. You may however, measure the distance to each of the apex points above you from each of the the three points you have on the ground plane. You will never measure angles, only distances between these 5 points, and the only distance you may not measure directly is between the two points hovering high in the air. How do you find this distance?

 

[mfb]

You can set up three equations with three unknowns for the positions of the apexes (individually) and solve, at least numerically. Once you have the 3D coordinates it's trivial to find the distance.

It's possible that there is some clever relation between the 10 distances (9 of them are known, so any relation will work).

Here is a shorter way to phrase the same problem by the way: There are 5 points in a three-dimensional space. You know 9 of the 10 distances between pairs of points. Find the 10th.

 

[Herbascious J]

You can set up three equations with three unknowns for the positions of the apexes (individually) and solve, at least numerically. Once you have the 3D coordinates it's trivial to find the distance.

It's possible that there is some clever relation between the 10 distances (9 of them are known, so any relation will work).

Here is a shorter way to phrase the same problem by the way: There are 5 points in a three-dimensional space. You know 9 of the 10 distances between pairs of points. Find the 10th.

Thank you for the restatement of the problem, that is much more forward and condensed. I was hoping that, yes, there would be some clever equation, that with enough crunching would fall out and I could simply plug in a set of measurements and always know the tenth distance. It seems that there should only be one or maybe two solutions (given one the pyramids might be inverted).