Let $A\subset {\mathbb{R}}^2$ be a finite set of size $n$, and let $\{d_1,\dots ,d_k\}$ be the set of distances determined by $A$. Let $f(d)$ be the multiplicity of $d$, that is, the number of ordered pairs from $A$ of distance $d$ apart. Is it true that $k=n-1$ and $\{f(d_i)\}=\{n-1,\dots ,1\}$ if and only if $A$ is a set of equidistant points on a line or a circle?