  We show that if a set of points in $\mathbb{C}^2$ lies on a family of $m$
concurrent lines, and if one of those lines contains more than $m-2$ points,
then there is a line passing through exactly two points of the set. The bound
$m-2$ in our result is optimal. Our main theorem resolves a conjecture of Frank
de Zeeuw, and generalizes a result of Kelly and Nwankpa.
