Vector equations and possible solutions

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I have two set vectors A, B, C, D and a, b, c, d.

Now we know (|A|-|a|),(|B|-|b|), (|C|-|c|), and (|D|-|d|)

We know combitions of (|A-B|-|a-b|), (|A-C|-|a-c|), (|A-D|-|a-d|), (|B-C|-|b-c|), (|B-D|-|b-c|) and (|C-D|-|c-d|) as well.

And all values are equal to zero. Is there any way to find out how many combinations of a, b, c, and d possible. At least mathematically so that all above relations hold.