Orthogonality related to Steiner and Pappus chains

Abstract

The focus of this paper is on the study of specific circle formations known as orthogonal Pappus chains and the related incidence results that involve points of tangency between the circles in the construction. Using circle inversion as a key tool, it is shown that these chains give rise to new families of circles whose centres lie on a conic determined by the configuration. This paper also explores similar questions for Steiner chains. Although an orthogonality condition cannot be formulated for them, we prove that when two Steiner chains share a common circle and admit an inversion sending the corresponding bounding pairs to concentric circles, several associated families of circles again have their centres lying on conics. Explicit examples are provided, exhibiting these analogous incidence phenomena and suggesting the presence of rich conic structures within Steiner chain configurations.