Theorem

Cayley-Bacharach Theorem

Geometry

The Cayley-Bacharach Theorem states that if two cubic curves in the plane intersect in exactly nine points, then any other cubic curve passing through eight of those nine points automatically passes through the ninth as well. Named for Arthur Cayley and Isaak Bacharach, it is a classical result of algebraic geometry with many variants relating the points where families of curves intersect, and it underlies facts such as the associativity of the elliptic curve group law.

Facts

Statement

If two cubics in the projective plane meet in nine different points, every cubic through any eight of them also passes through the ninth. 1

Classification

Statement Form

Characterization Theorem 1

Connections

Has Statement Form

Characterization

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Proved By

Arthur Cayley

Sources

1. Cayley-Bacharach theorem (Wikipedia)

Introduction

Quote, Introduction

Then every cubic that passes through any eight of the points also passes through the ninth point.

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Cayley-Bacharach Theorem

Has Statement Form

Proved By