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Comment by Sniffnoy | Hacker News Reader

I think what's going on here is that you've misunderstood the theorem's hypothesis. The hypothesis isn't that no three of the points are collinear; rather, it's the weaker statement that there isn't any one single line that all the points lie on. It's true that with your version of the hypothesis the theorem would be trivial; but with the actual hypothesis it is is nontrivial.

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> hypothesis

I think you meant premise?

The thing that you assume in a theorem is usually called a hypothesis.

looks like both words are valid, TIL!

>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"

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).

>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.

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.

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.

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.

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.

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