Chain Theorems of Lines Circles and Planes

Abstract

In this paper we will prove Clifford chain theorem for general lines in the plane by using real cross ratio lemma. We will then discuss the Clifford chain theorem for degenerate lines and will obtain 6 types of Clifford figures with examples. We will also find that the Clifford chain theorem fails for some cases and we will show some examples. At the end we will obtain three similar chain theorems for circles and planes by applying Clifford chain theorem for general lines in the plane under the transformations of stereographic projection and circle inversion. These three chain theorems are (1): chain theorem for general circles on the sphere, (2): chain theorem for general circles in the plane and (3): chain theorem for general planes in the space. We will also prove Miquel's pentagon theorem by applying Clifford chain theorem for four general lines.