Amanda Karlberg :Inversionsgeometri

Abstract:

This paper gives an introduction to inversive geometry. Initially, we introduce the basics of inversive geometry, including the definition and rules of inversion and how the transformation works for points, lines and circles. By using inversive geometry we then solve three different geometric problems. First we deal with Steiner’s porism, which states that wether a Steiner chain is open or closed does not depend upon where it starts. By inverting the Steiner chain in a specific circle of inversion, we easily can prove the porism. The second problem is related to an arbelos, a shoemaker’s knife, with an inscribed chain of circles. In this figure, the vertical distance between the base line and the center of a circle in the chain is determined in a certain way, and by inversion we can show how. The last one is called Apollonius’ problem and consist of three given circles, to whom we construct circles tangent