A Sylvester-Gallai result for concurrent lines in the complex plane

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.
Comment: 16 pages, 6 figures; several revisions added for overall clarity