Školjka

%V0 Let $k$ and $n$ be integers with $0\le k\le n - 2$. Consider a set $L$ of $n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $I$ the set of intersections of lines in $L$. Let $O$ be a point in the plane not lying on any line of $L$. A point $X\in I$ is colored red if the open line segment $OX$ intersects at most $k$ lines in $L$. Prove that $I$ contains at least $\dfrac{1}{2}(k + 1)(k + 2)$ red points. Proposed by Gerhard Woeginger, Netherlands