I propose a conjectured variant of Cayley-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 of degree $d$ that passes through any $\frac{d^2+3d}{2}-1$ of the points also passes through the $\frac{d^2+3d}{2}$ th point.