Does The Monge Theorem Apply To Some Non-Euclidean Geometries ?

Abstract

In geometry, Monge's theorem states that for any three nonoverlapping circles of distinct radii in the two dimensional analytical plane equipped with the Euclidean metric, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear. So, it is clearly observed that Monge's theorem is an application of Desargues' theorem. Our main motivation in this study is to show whether Monge theorem is still valid even if the plane is equipped with the metrics alpha and Lp.