The Cayley–Bacharach theorem is a result in algebraic geometry that deals with the intersection of divisors on a projective space. It is particularly relevant in the study of linear systems of divisors and their properties. In its classical form, the theorem states the following: Let \( C \) be a non-singular irreducible curve of degree \( d \) in the projective plane \( \mathbb{P}^2 \).
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In its classical nine-point cubic form, the Cayley-Bacharach theorem says that a cubic through eight of the nine distinct intersections of two plane cubics must contain the ninth. The eight-point cubic completion for two triples of lines is the elementary case needed for triangular-strip propagation in incidence geometry. The phenomenon explains why interpolation conditions at such a complete intersection are dependent.