Generalization of the Apollonius Circles

TL;DR

This paper explores the properties and generalizations of Apollonius circles in triangle geometry, demonstrating their coaxial nature under various configurations and transformations such as inversion and circumconics.

Contribution

It extends the classical Apollonius circle results to more general configurations, including arbitrary points and transformations like inversion, revealing new coaxial circle properties.

Findings

Apollonius circles are coaxal in classical and generalized settings.

Inversion in the incircle preserves coaxiality and introduces new circle centers.

Generalized circumconics relate to arbitrary points and their isogonal conjugates.

Abstract

The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are coaxal. Similarly defined three circles remain coaxal, when the circumcevian triangle is defined with respect to any point on the triangle circumconic through the incenter and symmedian point. Inversion in the incircle of the reference triangle carries these three coaxal circles into coaxal circles, each passing through a vertex of the inverted triangle and centered on the opposite sideline, at the intersection of the orthotransversal with respect to a point on the Euler line of the inverted triangle. A similar circumconic exists in a more general configuration, when the cevian triangle is defined with respect to an arbitrary point, passing through this arbitrary point and isogonal conjugate of its complement.