A Study on Circle Cross-Ratio in Möbius Geometry of Circles

In this paper, we study an invariant in the Möbius geometry of circles that is analogous to the cross-ratio in projective geometry. We refer to this invariant as the circle cross-ratio . We explore various properties of the circle cross-ratio that are analogous to those of the cross-ratio in projective geometry. We examine circle cross-ratio for the circles that lie in a pencil of circles. We present necessary and sufficient conditions under which the circle cross-ratio is positive or negative. Furthermore, for each real number r , we identify circles for which the circle cross-ratio equals r , and provide geometric interpretations of these circles. Building on these results, we finally show that in a hyperbolic bundle of circles, for each real number r , there exist circles whose circle cross-ratio is r .