The Area Method - A Recapitulation

The area method for Euclidean constructive geometry was pro posed by Chou, Gao and Zhang in the early 1990’s. The method can efficiently p rove many non-trivial geometry theorems and is one of the most interesting and most su cce sful methods for automated theorem proving in geometry. The method produces proo fs that are often very concise and human-readable. In this paper, we provide a first complete presentation of the method. We provide both algorithmic and implementation details that were omitted i n the original presentations. We also give a variant of Chou, Gao and Zhang’s axiom system. Bas ed on this axiom system, we proved formally all the lemmas needed by the method and its soundness using the Coq proof assistant. To our knowledge, apart from the original implementation by the authors who first proposed the method, there are only three implementations more . Although the basic idea of the method is simple, implementing it is a very challenging task because of a number of details that has to be dealt with. With the description of the method g iven in this paper, implementing the method should be still complex, but a straightforwar d t sk. In the paper we describe all these implementations and also some of their applicatio ns.