The Area Method

Abstract

The area method for Euclidean constructive geometry was proposed by Chou, Gao and Zhang in the early 1990's. The method can efficiently prove many non-trivial geometry theorems and is one of the most interesting and most successful methods for automated theorem proving in geometry. The method produces proofs 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 in the original presentations. We also give a variant of Chou, Gao and Zhang's axiom system. Based 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 given in this paper, implementing the method should be still complex, but a straightforward task. In the paper we describe all these implementations and also some of their applications. 490 P. Janičić et al.

Automating the Generation of High School Geometry Proofs using Prolog in an Educational Context

“…Practical applications include the recent integration of a deduction engine in GeoGebra [1], which is based on the internal representation of geometrical elements in complex numbers inside Ge-oGebra. Other examples include the systems based on the area method [7,11], the full-angle method [6], and many others. These systems seldom provide readable proofs, and when they do, they are far from what a high-school student would write.…”

Towards a Geometry Automated Provers Competition

“…For the preliminary run of GASC, during ThEdu'19, 2 the GCLC set of provers and some of the provers in GeoGebra's portfolio prover were selected [5,19,20]. The criteria used was: availability, reliability, and the possibility of running them in the command line (i.e.…”

“…It has a a built-in geometry theorem prover that can automatically prove a range of complex problems. The GATP module implements the Area Method, the Wu's method and the Gröbner Basis Method [18,19]. The implemented GATP can be called from inside the gclc-workbench (Linux) or WinGCLC (MS-Windows) DGS tools, but can be also used in an independent way.…”

“…CoqAM The formalisation of the area method using the proof assistant Coq was done by implementing the decision procedure as a Coq tactic and formalising all theorems needed by the method. The implementation guarantee the soundness of the method implementation, i.e., the proofs generated by the tactic are always correct [19,23].…”

“…In order to combine the geometric reasoning of synthetic methods and the efficiency of algebraic methods, some approaches, such as the area method and the the full-angle method, represent geometric knowledge in a form of expressions with respect to geometric invariants. These methods are broad-scoped, efficient and capable of producing geometric proofs [10,11,19].…”