# 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 |
---|---|

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 |