Cayley–Bacharach theorem

For two plane curves of degrees m and n meeting in mn points, every curve of degree m+n−3 through all but one intersection point also passes through the last, under the theorem’s intersection hypotheses.

Complete intersection: A variety defined by as many equations as its codimension, satisfying the expected dimension condition. The curve pair forms a complete intersection whose special point property the theorem describes.
Algebraic curve: A one-dimensional algebraic variety, usually defined by polynomial equations. The theorem concerns how two such plane curves intersect.
Arthur Cayley: An English mathematician who made foundational contributions to algebra, invariant theory, and geometry. Cayley’s work helped establish the result now associated with his name.
Complete quadrilateral: The six-point configuration formed by four lines in general position and their pairwise intersections. Its diagonal-point configuration gives a classical geometric setting for Cayley–Bacharach reasoning.
Polynomial interpolation: The construction of a polynomial satisfying prescribed values at specified points. Ordinary interpolation suggests independent point conditions; Cayley–Bacharach forces one condition to be dependent.
Projective plane: The geometric space of lines through the origin in a three-dimensional vector space, containing the ordinary affine plane as a chart. The theorem’s clean intersection statement is naturally formulated for projective plane curves.
Degree of a plane curve: The degree of a plane curve is the degree of its defining homogeneous polynomial. The degrees m and n determine both the intersection count and the testing degree.
Isaak Bacharach: A German mathematician known for work on algebraic geometry and the theorem bearing his name. Bacharach independently developed the geometric result that completed its standard eponym.
Nine-point theorem: A projective-plane result asserting a dependence among cubic curves through nine points arising from two cubics. The cubic case makes the theorem’s point-dependence concrete.
General position: A condition that geometric objects avoid special incidences or degeneracies. The theorem exploits a structured complete intersection rather than an arbitrary point set.

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Poncelet's closure theoremRelated: Algebraic-geometric arguments related to closure connect with incidence principles of this kind.