Find direction, angle or co-ord of unknown vertices using only distance?

Question

My current issue is that I have a triangle, where I know all the line distances as well as an origin coordinate.

Is there any way I can then gain the coordinates of the other vertices with this information?

If not, what is the minimum information needed to perform these calculations?

Thanks for your time,

M

Answer 1

The distances together with a point $P$ can, at most, determine the system up to rotations about $P$ and reflections about lines through $P$. For example, the distances are the same in each of the diagrams below:

If we have another fixed point, it still won't be enough, since we can still flip the positions about the line through those two points. For example, the distances are the same in each of the diagrams below:

Knowing the coordinates of three colinear points is also not sufficient, since again we can flip and preserve the distances:

(This may not apply if the points are assumed to be in general position.)

However, it is sufficient to know, in addition to the distances, the coordinates of three non-colinear points. Once those are placed, the distances to any other point is uniquely determined by their lengths to the three fixed points.