I propose a conjecture of variant  Caley-Bacharach's theorem

I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know some more information for the conjecture: 

**Conjecture:** Assume that two curves $C_1$ and $C_2$ in the projective plane meet in $\frac{d^2+3d}{2}$  (different) points, as they do in general over an algebraically closed field. Then every curve that passes through any  $\frac{d^2+3d}{2}-1$ of the points also passes through the  $\frac{d^2+3d}{2}$ th point.