In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane . The original form states: Assume that two cubics and in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes through any eight of the points also passes through the ninth point. A more intrinsic form of the Cayley–Bacharach theorem reads as follows: Every cubic curve over an algebraically closed field that passes through a given set of eight points also passes through (counting multiplicities) a ninth point which depends only on . A related result on conics was first proved by the French geometer Michel Chasles and later generalized to cubics by Arthur Cayley and Isaak Bacharach. Details If seven of the points lie on a conic, then the ninth point can be chosen on that conic, since will always contain the whole conic on account of Bézout's theorem. In other cases, we have the following. If no seven points out of are co-conic, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) (with multiplicity for double points) has dimension two. In that case, every cubic through also passes through the intersection of any two different cubics through , which has at least nine points (over the algebraic closure) on account of Bézout's theorem. These points cannot be covered by only, which gives us . Since degenerate conics are a union of at most two lines, there are always four out of seven points on a degenerate conic that are collinear. Consequently: If no seven points out of lie on a non-degenerate conic, and no four points out of lie on a line, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) has dimension two. On the other hand, assume are collinear and no seven points out of are co-conic. Then no five points of and no three points of are collinear. Since will always contain the whole line through on account of Bézout's theorem, the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) is isomorphic to the vector space of quadratic homogeneous polynomials that vanish (the affine cones of) , which has dimension two. Although the sets of conditions for both dimension two results are different, they