Sylvester-Gallai theorem

The Sylvester-Gallai theorem states that a finite noncollinear set of points in the real Euclidean plane determines a line containing exactly two of its points. Finiteness, noncollinearity and the real field matter. A useful proof chooses the smallest positive distance between a point $P$ and a line $L$ determined by the set. If $L$ contains at least three points, two of them, $A,B$, lie on the same closed side of the perpendicular foot, with $A$ nearer. The distance from $A$ to $PB$ is the original altitude multiplied by $AB/PB<1$, contradicting minimality. Thus $L$ contains exactly two points. Related statements over complex projective space require a different conclusion.^[ref]