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]:
     Fig4 http://cs.smith.edu/%7Eorourke/MathOverflow/FourPointConfig.png

It is shown though (also Kemper, I believe) that for most configurations the set of distances determine the configuration (which is probably intuitively expected).

The example I have mentioned answers you question, but actually it would be natural if in your question you also require that the distances come with their multiplicities.

There exists examples of different point configurations in $\mathbb{R}^2$ 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]:
     

It is shown though (also Kemper, I believe) that for most configurations the set of distances determine the configuration (which is probably intuitively expected).

The example I have mentioned answers your question, but actually it would be natural if in your question you also require that the distances come with their multiplicities.

There exists examples of different point configurations in $R^2$ having the same the set (but different matrices of!) distanced. 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 the page 5 to see the picture.

It is shown though (also Kemper, I believe) that for most configurations the set of distances determine the configuration (which is probably intuitively expected).

The example I have mentioned answers you question, but actually it would be natural if in your question you also require that the distances come with their multiplicities.

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]:
     Fig4 http://cs.smith.edu/%7Eorourke/MathOverflow/FourPointConfig.png

It is shown though (also Kemper, I believe) that for most configurations the set of distances determine the configuration (which is probably intuitively expected).

The example I have mentioned answers you question, but actually it would be natural if in your question you also require that the distances come with their multiplicities.