> Every finite set of points in the Euclidean plane that is not collinear has a line that passes through exactly two of the points.
I can't make out the point here (no pun). Of course a line can pass through any two points. It could pass through three if those points were collinear but the statement says they're not. So what is the new fact?
glimshe 35 minutes ago [-]
Try to come up with a set non-colinear points where NO line passes through two and ONLY TWO points and you'll see the value of the statement.
You may think "I'm sure I can arrange these points in a way where EVERY line will cross three or more points" but you will fail if you try unless ALL points are colinear.
scythe 6 minutes ago [-]
This is true for finite sets. For infinite sets, the Sierpinski triangle is a counterexample.
chenb4425 49 minutes ago [-]
It’s that the line passes through exactly two points, which if you think about it is not exactly obvious.
math_loser 47 minutes ago [-]
> So what is the new fact?
For all arbitrarily sized (but finite) sets of not collinear points, there's always a line that passes through exactly two points in the set.
tzs 38 minutes ago [-]
It can help to think about theorems like this by restating them as a puzzle asking for a counterexample.
Given N points, N > 2, can you arrange them in a Euclidean plane so that (1) they are not all on the same line, and (2) every line that goes through two of the points must also go through at least one more of the points?
The theorem says that you cannot do this.
Nail2680 12 minutes ago [-]
I might be too stupid to understand why this is interesting and useful. If it helps I am a working physicist, and a lot of pure math is lost on me. I think I followed this, but I don't know why one would care or this would be interesting.
emil-lp 2 hours ago [-]
Futility closet is fantastic!
Rendered at 03:37:08 GMT+0000 (Coordinated Universal Time) with Vercel.
I can't make out the point here (no pun). Of course a line can pass through any two points. It could pass through three if those points were collinear but the statement says they're not. So what is the new fact?
You may think "I'm sure I can arrange these points in a way where EVERY line will cross three or more points" but you will fail if you try unless ALL points are colinear.
For all arbitrarily sized (but finite) sets of not collinear points, there's always a line that passes through exactly two points in the set.
Given N points, N > 2, can you arrange them in a Euclidean plane so that (1) they are not all on the same line, and (2) every line that goes through two of the points must also go through at least one more of the points?
The theorem says that you cannot do this.