A General Theory of Affine Geometry and Two Extensions of Hoehn’s Theorem

Abstract

This thesis sets up a framework for planar affine geometry with an algebraic approach. We start with points and affine combination as the only operation on points. This allows us to develop objects such as lines and vectors, and operations such as 1-combination and 0-combination; and derive basic properties such as parallelism and signed area. Such foundation is then further enhanced with barycentric coordinates of lines to elevate lines as a subject of interest in triangle geometry. A closer look at Ceva’s and Menelaus’ theorems leads to the finding of Ceva/Menelaus triples. The area principle and parametrisation of a non-degenerate general conic are then introduced as tools for studies in affine geometry. As a direct application, Hoehn’s theorem is further investigated for its general application on general n-gons, resulting in the n-gon Proportion Product and Conic n-gon Proportions theorems.