proof of Erdös-Anning Theorem

Let $A, B$ and $C$ be three non-collinear points. For an additional point $P$ consider the triangle $A B P$ . By using the triangle inequality for the sides $P B$ and $P A$ we find $-|AB|\\leq|PB|-|PA|\\leq|AB|$ . Likewise, for triangle $B C P$ we get $-|BC|\\leq|PB|-|PC|\\leq|BC|$ . 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.

单词	ProofOfErdosAnningTheorem
释义	proof of Erdös-Anning Theorem Let $A, B$ and $C$ be three non-collinear points. For an additional point $P$ consider the triangle $A B P$ . By using the triangle inequality for the sides $P B$ and $P A$ we find $-\|AB\|\\leq\|PB\|-\|PA\|\\leq\|AB\|$ . Likewise, for triangle $B C P$ we get $-\|BC\|\\leq\|PB\|-\|PC\|\\leq\|BC\|$ . 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.
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