3D Pyramid Puzzle, Vector from hypotenuse lengths

This has currently got me stumped and im hoping someone may know the formula for working this out, its quit hard to explain but ill try my best. Its to take data from a winch automation system where i am given the wire lengths and need to know where they join/intersect in 3d space to represent the actual object.

Picture this:

Four steel wires hanging from the corners of a roof of a room which is 4m x 4m.
Where the four wires come together an object is hanging off this point and by adjusting the lengths of each wire we can move the object hanging off them in 3D space within the confines of the “room x,y,z” which we theoretically know.

if we know that the base points for each wire is at each corner of a square which is 4m x 4m.
AND we know the length of each wire, does anyone know a way or a formula to interpolate these values to give the position of the apex in x,y,z values.

Im absolutely certain there is a way but i may be searching in the wrong direction, its quite an exciting challenge but i am currently stuck! Any suggestions pointers or tips would be much appreciated!

P.S, i apologise if i have completely missed a node which does this in 3d in vvvv, im only picking this up again after a while!



here is one idea, the inverse problem is quite easy:

L1 = |X-P1|
L2 = |X-P2|
L3 = |X-P3|
L4 = |X-P4|

you could write out the formulas and solve for X = (x,y,z). the problem will be, that you have 3 unknown and 4 equations, but that should be no problem, because you can calculate the fourth length always from the 3 others. so maybe solve the problem first for 3 lenghts and then go on…

Hi Tonfilm, thanks, im a but unsure of your formula, i think ive worked it out, it want easy, a bit of thinking inside the pyramid, ill publish it here when i find a sensible way to note it down!