A theorem about vectors in R² and an algebraic proof of a conjecture of Erdős and Purdy

Let V be a set of n vectors in R². Assume that for every distinct v, v^′ and v^′′ in V, the vectors v + v^′ and v + v^′′ are linearly independent. We show that in such a case the set of vectors {v + v^′ | v, v^′ ∈ V, v ≠ v^′} contains at least n vectors every two of which are linearly independent, unless n = 2, n = 4, n = 6, or n ≥ 8 is even and O, the origin is in V . In the latter case the other n − 1 vectors are (up to a linear transformation) the set of vertices of a regular (n − 1)-gon centered at O. We use this result to provide a short algebraic proof of an old conjecture of Erdős and Purdy: Let P be a set of n points in general position in the plane. Suppose that R is a set of red points disjoint from P such that every line determined by P passes through a point in R. Then |R| ≥ n, unless n = 2 or n = 4.