proof of Erdös-Anning Theorem
Let and be three non-collinear points. For an additional point consider the triangle . By using the triangle inequality for the sides and we find . Likewise, for triangle we get . Geometrically, this means the point lies on two hyperbola with and or and respectively as foci. Since all the lengths involved here are by assumption integer there are only possibilities for and possibilities for . These hyperbola are distinct since they don’t have the same major axis. So for each pair of hyperbola we can have at most points of intersection and there can be no more than points satisfying the conditions.
|Title||proof of Erdös-Anning Theorem|
|Date of creation||2013-03-22 13:19:11|
|Last modified on||2013-03-22 13:19:11|
|Last modified by||lieven (1075)|