Sylvester’s theorem

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


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 distanceMathworldPlanetmath d(A,l) between them is minimalPlanetmathPlanetmath. 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 perpendicularMathworldPlanetmathPlanetmathPlanetmath 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
Canonical name SylvestersTheorem
Date of creation 2013-03-22 13:59:36
Last modified on 2013-03-22 13:59:36
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 5
Author bbukh (348)
Entry type Theorem
Classification msc 52C35
Classification msc 51M04