# proof of Erdös-Anning Theorem

Let $A,B$ and $C$ be three non-collinear points. For an additional point $P$ consider the triangle^{} $ABP$. By using the triangle inequality for the sides $PB$ and $PA$ we find $-|AB|\le |PB|-|PA|\le |AB|$. Likewise, for triangle $BCP$ we get $-|BC|\le |PB|-|PC|\le |BC|$. Geometrically, this means the point $P$ lies on two hyperbola^{} with $A$ and $B$ or $B$ and $C$ respectively as foci. Since all the lengths involved here are by assumption^{} integer there are only $2|AB|+1$ possibilities for $|PB|-|PA|$ and $2|BC|+1$ possibilities for $|PB|-|PC|$. These hyperbola are distinct since they don’t have the same major axis. So for each pair of hyperbola we can have at most $4$ points of intersection^{} and there can be no more than $4(2|AB|+1)(2|BC|+1)$ points satisfying the conditions.

Title | proof of Erdös-Anning Theorem |
---|---|

Canonical name | ProofOfErdosAnningTheorem |

Date of creation | 2013-03-22 13:19:11 |

Last modified on | 2013-03-22 13:19:11 |

Owner | lieven (1075) |

Last modified by | lieven (1075) |

Numerical id | 4 |

Author | lieven (1075) |

Entry type | Proof |

Classification | msc 51-00 |