The full angle method on the OpenGeoProver

Geometry with its formal, logical and spatial properties is well suited to be taught in an environment that includes dynamic geometry software (DGSs), geometry automated theorem provers (GATPs) and repositories of geometric problems (RGPs). With the integration of those tools in a given learning environment the student is able to explore the built-in knowledge, but also to do new constructions, and to test new conjectures. In such an environment the student can visualise geometric objects and link the formal, axiomatic, nature of geometry with its standard models and corresponding illustrations, e.g., Euclidean Geometry and the Cartesian model. With the help of a geometry automated theorem prover it is possible to check the correctness of the constructions, e.g. if two given lines are parallel, and also to make formal proofs of geometric conjectures. Automated theorem proving in geometry has two major lines of research: synthetic proof style and algebraic proof style [1, 13]. Algebraic proof style methods are based on reducing geometric properties to algebraic properties expressed in terms of Cartesian coordinates. These methods are usually very efficient, but the proofs they produce do not reflect the geometrical nature of the problem and they give only yes or no conclusion. Synthetic methods attempt to automate traditional geometry proof methods, producing human-readable proofs but they are inefficient [7]. The area method and the full-angle method are two semi-synthetic methods providing traditional (not coordinate-based), human-readable proofs. The proofs are expressed in terms of higher-level geometric lemmas and expression simplifications. The main idea of the method is to express the hypotheses of a theorem using a set of constructive statements, each of them introducing a new point or a new line, and to express the conclusion by an equality of expressions in some geometric quantities (e.g. the signed area of a triangle), without referring to Cartesian coordinates. The proof is then based on eliminating (in reverse order) the points and/or lines introduced before, using a set of appropriate lemmas for that purpose. After eliminating all introduced elements, the current goal becomes an equality between two expressions in quantities over independent points. If it is trivially true, then the original conjecture was proved valid, if it is trivially false, then the conjecture was proved invalid, otherwise, the conjecture has been neither proved nor disproved [3, 4, 5]. The following simple example briefly illustrates some key features of this type of methods.