Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.
Nine Associated Points Theorem
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References
Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. The Seven Circles Theorem and Other New Theorems. London: 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 Web Resource. https://mathworld.wolfram.com/NineAssociatedPointsTheorem.html