Balanced configurations of 2n+1 plane vectors

A plane configuration {v_1,...,v_m} of vectors in {\mathbb R}^2 is said to be balanced if for any index i, the set of the det(v_i,v_j) for j\neq i is symmetric around the origin. A plane configuration is said to be uniform if every pair of vectors is linearly independent. E. Cattani and A. Dickenstein conjectured that any uniform balanced configuration is GL_2({\mathbb R})-equivalent to a regular (2n+1)-gon. In this note, we prove this conjecture.