Let $A_1, A_2,\ldots, A_n$ be $n$ distinct points in the plane. For every point $A_i$, let $D_i$ be the set consisting of the distances, under the usual Euclidean metric, from $A_i$ to all the other points. Regard each $D_i$ as a multiset with exactly $n-1$ elements. This is needed since it may be that some of the distances are the same.

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

Is it true that $A_1, A_2, \ldots A_n$ are the vertices of a regular polygon?

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?