A Proof of Seven Circles Theorem using Hyperbolic Geometry

Abstract

In this paper, we explore a non-conventional proof of the Seven Circles Theorem using several concepts from hyperbolic geometry. We attempt to represent the picture, claimed by the statement, in the Klein model—followed by the Poincar´e’s hyperbolic disk model of hyperbolic space—in order to analyze the claim. We consider an ideal hexagon to have been formed by the points of intersection of each of the six inner circles and the ideal boundary. We then assume that there exists a non-ideal hyperbolic triangle that is formed as a result of intersections between the three main diagonals of the hexagon. We then go on to contradict this claim by proving that the area of the non-ideal triangle is zero.