Generalization of Apollonius Circle

TL;DR

This paper extends the classical Apollonius circle concept from points to circles, showing that the locus of points with a constant ratio of power with respect to two circles forms a circle, thus broadening its geometric applications.

Contribution

The paper generalizes the Apollonius circle definition from points to circles and proves the locus of points with a constant power ratio is also a circle, expanding classical geometric results.

Findings

Locus of points with constant power ratio to two circles is a circle.

Generalization of Apollonius circle from points to circles.

Extended classical properties 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.