Cayley–Bacharach Theorem

The Cayley-Bacharach theorem states that if two cubics in the projective plane meet in nine (different) points, then every cubic that passes through eight of them also passes through the ninth.1 The fundamental reason behind this is that polynomial rings over a field are Gorenstein.2

Three Conics Theorem3

The three conics theorem is a particular case of the Cayley-Bacharach theorem when each cubic is a union of a straight line and a conic. This viewpoint provides a high-level proof and establishes a clear relationship with Pascal’s theorem.

Five Conics Theorem4

The five conics theorem is also straightforward. Using the naming from the paper, we prove that \(P_1Q_1, P_3Q_3,\) and \(IJ\) are concurrent, by the application of the three conics theorem to \(S_1, S_2, S\). Then we obtain the desired result using the converse three conics theorem to \(S_3, S_4, S\).

  1. Wikipedia, Cayley–Bacharach theorem. ↩

  2. Eisenbud, David, Mark Green, and Joe Harris. “Cayley-Bacharach theorems and conjectures.” ↩

  3. Weisstein, Eric W. “Three Conics Theorem.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/ThreeConicsTheorem.html ↩

  4. Tran Thu Le and Kien Trung Nguyen. “The Five Conics Problem.” ↩