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The Sylvester–Gallai Theorem - Futility Closet

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The Sylvester–Gallai Theorem - Futility Closet

Every finite set of points in the Euclidean plane that is not collinear has a line that passes through exactly two of the points. This proof is by Michigan State University mathematician Leroy Milton Kelly. Consider a set S of points that aren’t all collinear, and define a connecting line to be a line that contains at least two of these points. There must be some point P and connecting line ℓ that are closer together than any other point-line pair in the set. Kelly now proves that ℓ contains only two of the points in S. Assume that this…

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