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Let $A_1, A_2,\ldots, A_n$ be $n$ distinct points in the plane. For every $1\le i\le n$, let $D_i$ be the sum of the distances from point $A_i$ to all the other points.

Suppose that $D_i=D_j$ for every $1\le i< j \le n$.

Is it true that $A_1, A_2, \ldots A_n$ must be concyclic?

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    $\begingroup$ Not necessarily: this will hold for any shape with a transitive group of isometries, which includes for example non-square rectangles. $\endgroup$
    – Ben Barber
    Commented Nov 19, 2015 at 14:42
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    $\begingroup$ I'm voting to close this question as off-topic because it's not research level (and answered in the comments). $\endgroup$
    – Igor Rivin
    Commented Nov 19, 2015 at 14:48
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    $\begingroup$ Note: the original question was put on hold. The comments above as well as @Joseph's answer are replying to the old question. The question was changed after a related question by the same user was answered (here: mathoverflow.net/questions/224037/…). I voted to reopen this question. $\endgroup$
    – Yoav Kallus
    Commented Nov 20, 2015 at 20:53

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This is, in general, a difficult problem, addressed in the paper below. It goes under the phrase: the beltway reconstruction problem.

Lemke, Paul, Steven S. Skiena, and Warren D. Smith. "Reconstructing sets from interpoint distances." Discrete & Computational Geometry. Springer Berlin Heidelberg, 2003. 597-631.

There is some info in an earlier MO question. Sets with nonunique reconstructions are called homometric. Here is a snippet from the above paper, addressing a property of regular polygons:

         
          (From p.335 of conference version.)

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