I propose a conjecture variant of [Cayley-Bacharach's theorem](https://en.wikipedia.org/wiki/Cayley%E2%80%93Bacharach_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.



A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem
I do not understand how your $d$ is related to the degrees of $C\_1$ and $C\_2$. Anyway, many variants and generalizations of the classical Cayley-Bacharach for cubics are known. One of them is the statement below, whose proof can be found in the beautiful paper by Eisenbud, Green and Harris 


*[Cayley-Bacharach Theorems and conjectures](http://www.ams.org/journals/bull/1996-33-03/S0273-0979-96-00666-0/)*, Bull. Amer. Math. Soc. **33** (1996).



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> **Theorem (Generalized Cayley-Bacharach).** Let $C\_1$ and $C\_2$ be two curves in $\mathbb{P}^2$ of degrees $d$ and $e$, respectively. Assume that they intersect in a collection of $d \cdot e$ distinct points $\Gamma = \{p\_1, \ldots, p\_{de}\}$. If $C \subset \mathbb{P}^2$ is a curve of degree $d+e-3$ containing all but one point of $\Gamma$, then $C$ contains all of $\Gamma$. 
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