Comment by glimshe
4 hours 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.
You can always find a line that passes between two points. Why would you even try to find a line that doesn’t pass between two points? What is the difficult part here.
Find one that bridges only two points, no more. Unless ofc you're given a configuration that's obviously impossible: all of them in a line.
This is true for finite sets. For infinite sets, the Sierpinski triangle is a counterexample.
> the Sierpinski triangle is a counterexample
How so? It's bounded by the large initial triangle. The line containing any two of the vertices doesn't intersect any other point.