I try the method of induction on it, but I fail at the last step. I assume that the statement is true for all planes with less than n points. Then if I add one more point to the plane so that it is not collinear to the line with exaclty two points on it, the statement is true for the plane with n points. However, if the new point is added on the line with exactly two points, how can we make sure that there is still a line passing through exactly two points?
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2Previous related questions: https://math.stackexchange.com/questions/3772820/prove-sylvester-gallai-theorem-using-combinatorics and https://math.stackexchange.com/questions/754430/sylvester-gallai-problem and https://math.stackexchange.com/questions/2608906/proof-of-sylvester-gallai-theorem-using-induction and https://math.stackexchange.com/questions/531070/examples-of-famous-problems-resolved-easily/532096#532096 and https://math.stackexchange.com/questions/3259939/are-there-finitely-many-points-in-mathbb-r2-that-satisfy-this-condition/3260247#3260247 and many others. – Gerry Myerson Aug 26 '20 at 13:21
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This is called the Sylvester-Gallai Theorem. You can find many proofs on the internet, including in the Wikipedia article:
https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem
See the following notes for a nice and slick proof: http://web.stanford.edu/~yuvalwig/math/teaching/WhatsThePoint.pdf
halrankard
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1The terminology is "ordinary line" (rather than "ordinary of a line"). This is defined at the beginning of the article; it's a line that goes through just two points in the collection. – halrankard Aug 26 '20 at 12:43
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