= Sylvester-Gallai theorem
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= Sylvester-Gallai
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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.<Sylvester-Gallai Theorems for Complex Numbers and Quaternions by Noam Elkies and others>[introduction]{ref}
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