MathNet 2011 Question 69

For every $n \ge 3$, determine all the configurations of $n$ distinct points $X_1, X_2, \dots, X_n$ in the plane, with the property that for any pair of distinct points $X_i, X_j$ there exists a permutation $\sigma$ of the integers $\{1, \dots, n\}$, such that $d(X_i, X_k) = d(X_j, X_{\sigma(k)})$ for all $1 \le k \le n$. (We write $d(X, Y)$ to denote the distance between points $X$ and $Y$.)

(United Kingdom) Luke Betts