[HN Gopher] The Sylvester-Gallai Theorem
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       The Sylvester-Gallai Theorem
        
       Author : surprisetalk
       Score  : 23 points
       Date   : 2026-07-31 14:00 UTC (6 days ago)
        
 (HTM) web link (www.futilitycloset.com)
 (TXT) w3m dump (www.futilitycloset.com)
        
       | emil-lp wrote:
       | Futility closet is fantastic!
        
       | hyperhello wrote:
       | > 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?
        
         | chenb4425 wrote:
         | It's that the line passes through _exactly_ two points, which
         | if you think about it is not exactly obvious.
        
         | math_loser wrote:
         | > 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 wrote:
         | 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.
        
         | glimshe wrote:
         | 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 wrote:
           | This is true for _finite_ sets. For infinite sets, the
           | Sierpinski triangle is a counterexample.
        
         | Sniffnoy wrote:
         | 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.
        
       | Nail2680 wrote:
       | 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.
        
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