In this paper we report on our recent study of Clifford algebra for geometric reasoning and its application to problems in computer vision. A general framework is presented for construction and representation of geometric objects with selected rewrite rules for simplification. It provides a mechanism suitable for devising methods and software tools for geometric reasoning and computation. The feasibility and efficiency of the approach are demonstrated by our preliminary experiments on automated theorem proving in plane Euclidean geometry. We also explain how non-commutative Grobner bases can be applied to geometric theorem proving. In addition to several well-known geometric theorems, two application examples from computer vision are given to illustrate the practical value of our approach.