Comment by agnishom
2 hours ago
That is not what was meant. Here is a better rephrasing:
Let X be a set of points not all of which are collinear. Then, there are two points a, b in X such that the line l passing through X only passes through a and b.
>Let X be a set of points not all of which are collinear. Then, there are two points a, b in X such that the line l passing through X only passes through a and b.
I don't see how this rephrasing changes anything. Of course there are two points a and b because again, the definition of the problem leads naturally, obviously, and definitionally to this result.
Not all points being collinear does NOT mean that all 3-tuples of points are non-collinear! The hypothesis of the theorem is the former. And what it proves is that there is at least one such 3-tuple.
The other thread above helped me. You can have as many collinear points as you want as long as at least one point in the set is non-collinear.
Consider a 3x3 grid. It satisfies this argument.