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 $x,y\in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$ and $y$. We show that $P\cup R$ must be contained in a cubic curve.