The nine associated points theorem is the Cayley-Bacharach theorem for cubic curves. It states that any cubic curve passing through eight of the nine distinct intersection points of two given cubic curves also passes through the ninth.
Nine Associated Points Theorem
See also
Cayley-Bacharach TheoremExplore with Wolfram|Alpha
References
Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. The Seven Circles Theorem and Other New Theorems. London, England: Stacey International, p. 15, 1974.Referenced on Wolfram|Alpha
Nine Associated Points TheoremCite this as:
Weisstein, Eric W. "Nine Associated Points Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NineAssociatedPointsTheorem.html