Generalization of Apollonius Circle

Abstract

Apollonius of Perga, showed that for two given points $A,B$ in the Euclidean plane and a positive real number $k\neq 1$, geometric locus of the points $X$ that satisfies the equation $|XA|=k|XB|$ is a circle. This circle is called Apollonius circle. In this paper we generalize the definition of the Apollonius circle for two given circles $\Gamma_1,\Gamma_2$ and we show that geometric locus of the points $X$ with the ratio of the power with respect to the circles $\Gamma_1,\Gamma_2$ is constant, is also a circle. Using this we generalize the definition of Apollonius Circle, and generalize some results about Apollonius Circle.