Apollonian sets in taxicab geometry

Abstract

Fix two points $p$ and $q$ in the plane and a positive number $k\ne 1$. A result credited to Apollonius of Perga states that the set of points $x$ that satisfy $d\left(x,p\right)∕d\left(x,q\right)=k$ forms a circle. In this paper we study the analogous set in taxicab geometry. We find that while Apollonian sets are not taxicab circles, more complicated Apollonian sets can be characterized in terms of simpler ones.