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?

Futility closet is fantastic!