  A balanced configuration of points on the sphere $S^2$ is a (finite) set of
points which are in equilibrium if they act on each other according any force
law dependent only on the distance between two points. The configuration is
additionally group-balanced if for each point in a configuration $\mathcal{C}$,
there is a symmetry of $\mathcal{C}$ fixing only that point and its antipode.
Leech showed that these definitions are equivalent on the sphere $S^2$ by
classifying all possible balanced configurations. On the other hand, Cohn,
Elkies, Kumar, and Sch\"urmann showed that for $n\geq 7,$ there are examples of
balanced configurations in $S^{n-1}$ which are not group balanced. They also
suggested extending the notion of balanced configurations to Euclidean space,
and conjectured that at least in the case of the plane, all discrete balanced
configurations in $\mathbb{R}^n$ are group-balanced. We verify a reformulation
of this conjecture by providing a complete classification of the balanced
configurations in $\mathbb{R}^2$ satisfying a certain minimal distance
property.
