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.