# Sylvester’s theorem

For every finite collection of non-collinear points in Euclidean space, there is a line that passes through exactly two of them.

###### Proof.

Consider all lines passing through two or more points in the collection. Since not all points lie on the same line, among pairs of points and lines that are non-incident we can find a point $A$ and a line $l$ such that the distance $d(A,l)$ between them is minimal. Suppose the line $l$ contained more than two points. Then at least two of them, say $B$ and $C$, would lie on the same side of the perpendicular from $p$ to $l$. But then either $d(AB,C)$ or $d(AC,B)$ would be smaller than the distance $d(A,l)$ which contradicts the minimality of $d(A,l)$. ∎

Title Sylvester’s theorem SylvestersTheorem 2013-03-22 13:59:36 2013-03-22 13:59:36 bbukh (348) bbukh (348) 5 bbukh (348) Theorem msc 52C35 msc 51M04