Given the pairwise distances between $n$ points, how can I find plausible coordinates for the points? If I have three points $A, B, C$, and I know the distances between $A$ and $B$, $B$ and $C$, and $A$ and $C$,
(1) How can I find (one possible value for) the coordinates of $A$, $B$, and $C$?
(2) If the distance function is a viable distance metric, must a solution always exist?
(3) How can I generalize this to $n$ points?

Because the distances do not change with translations, reflection and rotations, we can (even if we are in 3D) restrict to the $x,y$ plane, and fix $A=(0,0)$ , $B=(b_1,0)$, $C=(c_1,c_2)$
Then $b_1 = d_{AB}$. Also, $c_1^2 + c_2^2 = d_{AC}^2$ and $(c_1-b_1)^2 + c_2^2 = d_{BC}^2$
This gives: $$c_1 = \frac{d_{AC}^2 - d_{BC}^2 + d_{AB}^2}{2 d_{AB}}$$
and
$$ c_2^2 = d_{AC}^2 - c_1^2$$
which must be positive to have a solution (well, two). From this we get the general points by rotations, translations and reflections. I'm not sure if there is a more elegant or generalizable way.
