The Sylvester–Gallai Theorem

> 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?

>Every finite set of points in the Euclidean plane that is not collinear has a line that passes through exactly two of the points.

Isn't this a tautology?

The problem definition states that the set of points is in Euclidean space, which from Euclid's Axioms means we can draw a line between any two points. The set of points is defined to be not collinear, thus we cannot draw a line passing through more than two of them. This is just simple logic.

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.

Futility closet is fantastic!