3 ms·

>rather, it's the weaker statement that there isn't any one single line that all the points lie on ... of course there's no single line that all the points lie

>rather, it's the weaker statement that there isn't any one single line that all the points lie on ... of course there's no single line that all the points lie on. They've been defined to be non-collinear. Edit: can't reply because of HN's stupid rate-limit mechanism, but to this: >So the theorem proves that no matter which way you arrange any finite set of points, except for all on the same line, then you can always find a line with exactly two points. Of course you can. It's absolutely implied by the problem definition. My 9 year old could do this, given a ruler and a pencil, with 100% success rate. I absolutely do not believe this is a novel "theorem"

hyperhello 2mo ago
His statement helped me. It's not that every three points are non-collinear, it's that any three points are non-collinear. A set of points all lying on a line is the only exception; you can have every point lying on a line except for one, or two, or whatever you want. In a square grid of sixteen points, there are lots of sets of four collinear points for example, but not all sixteen, and that's what counts. So the theorem proves that no matter which way you arrange any finite set of points, except for all on the same line, then you can always find a line with exactly two points.
chenb4425 2mo ago
I think you are mistaking the fact that you can easily find an example satisfying the theorem’s statement with the proof that the statement is always true. Of course given any set of points that aren’t all on the same line, your nine your old could find a line passing through only two points. But could they explain to you why this is always possible, no matter the configuration of points? You can’t just say “I draw a line between two points and that’s it.” You must also explain why there isn’t a third point on the line, and why that line’s existence is guaranteed, which is not obvious (at least to me).
- stackghost 2mo ago
  >You must also explain why there isn’t a third point on the line, and why that line’s existence is guaranteed, which is not obvious (at least to me). There isn't a third point on the line you found because the problem stipulates that the set of points is not collinear.
  - chenb4425 2mo ago
    That's only assuming your set of points only contains three points. If you have four or more points, it's entirely possible that a line you draw between two points passes through a third. . ... If you're given the above set of points, it's obviously not collinear due to the point at the top, but if you draw a line through any of the bottom two points, it will hit a third.
  - h3lp 2mo ago
    So it says that not ALL points are on that single line. But it doesn't stipulate that THREE points can't be on the line---the theorem is about proving that you can find one that doesn't have the third point.
emil-lp 2mo ago
You don't understand the statement. A bit of mathematical maturity is needed, sometimes, to parse a theorem statement. In any finite set of points, either there is a line hitting all points, or there is a line hitting exactly 2 points. It's nontrivial to prove.

hyperhello 2mo ago

His statement helped me. It's not that every three points are non-collinear, it's that any three points are non-collinear. A set of points all lying on a line is the only exception; you can have every point lying on a line except for one, or two, or whatever you want. In a square grid of sixteen points, there are lots of sets of four collinear points for example, but not all sixteen, and that's what counts. So the theorem proves that no matter which way you arrange any finite set of points, except for all on the same line, then you can always find a line with exactly two points.

chenb4425 2mo ago

I think you are mistaking the fact that you can easily find an example satisfying the theorem’s statement with the proof that the statement is always true. Of course given any set of points that aren’t all on the same line, your nine your old could find a line passing through only two points. But could they explain to you why this is always possible, no matter the configuration of points? You can’t just say “I draw a line between two points and that’s it.” You must also explain why there isn’t a third point on the line, and why that line’s existence is guaranteed, which is not obvious (at least to me).

stackghost 2mo ago
>You must also explain why there isn’t a third point on the line, and why that line’s existence is guaranteed, which is not obvious (at least to me). There isn't a third point on the line you found because the problem stipulates that the set of points is not collinear.
- chenb4425 2mo ago
  That's only assuming your set of points only contains three points. If you have four or more points, it's entirely possible that a line you draw between two points passes through a third. . ... If you're given the above set of points, it's obviously not collinear due to the point at the top, but if you draw a line through any of the bottom two points, it will hit a third.
- h3lp 2mo ago
  So it says that not ALL points are on that single line. But it doesn't stipulate that THREE points can't be on the line---the theorem is about proving that you can find one that doesn't have the third point.

emil-lp 2mo ago

You don't understand the statement. A bit of mathematical maturity is needed, sometimes, to parse a theorem statement. In any finite set of points, either there is a line hitting all points, or there is a line hitting exactly 2 points. It's nontrivial to prove.