Is list of distances defining points uniquely ?

There is N points on a plane. Is is feasible to reproduce there relative location having only list of distances. Assuming that translation, rotation and mirror are allowed in the result. The list contains only distances between every pair of points, but not which points these are.

For simple triangle like A=(0,0) B=(1,0) C=(1,1) the distance are: |AB| = 1 |AC| = sqrt(2) |BC| = 1

The list would be: 1, 1, sqrt(2).

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See the earlier MO question, "Reconstructing an Euclidean point cloud from their pairwise distances" mathoverflow.net/questions/97611 , which refers to even earlier MO questions. As I said there: Short version: The problem is NP-hard; search under the phrase "distance geometry." – Joseph O'Rourke Jul 14 '12 at 12:10
Yes, I have seen that question. This version of the problem lacks the information about point for which distance is given. Isn't that a problem? – janst Jul 14 '12 at 13:08

There exists examples of different point configurations in $R^2$ having the same the set (but different matrices of!) distances. The simplest example contains 4 points and could be found in the paper of Boutin and Kemper, see http://arxiv.org/pdf/math/0304192v1.pdf -- scroll to page 5 to see the picture [Added by J.O'Rourke]: