Cayley-Bacharach Theorem

Let $X_1, X_2\subset\Bbb{P}^2$ be Cubic plane curves meeting in nine points $p_1$, ..., $p_9$. If $X\subset\Bbb{P}^2$ is any Cubic containing $p_1$, ..., $p_8$, then $X$ contains $p_9$ as well. It is related to Gorenstein Rings, and is a generalization of Pappus's Hexagon Theorem and Pascal's Theorem.

References

Eisenbud, D.; Green, M.; and Harris, J. ``Cayley-Bacharach Theorems and Conjectures.'' Bull. Amer. Math. Soc. 33, 295-324, 1996.