On Sets of n Points in General Position That Determine Lines That Can Be Pierced by n Points

Let P be a set of n points in general position in the plane. Let R be a set of n points disjoint from P such that for every x, y∈ P the line through x and y contains a point in R outside of the segment delimited by x and y. We show that P∪ R must be contained in cubic curve. This resolves a special case of a conjecture of Milićević. We use the same approach to solve a special case of a problem of Karasev related to a bipartite version of the above problem.