Automated Geometric Reasoning with Geometric Algebra: Theory and Practice

The field of Automated Theorem Proving was initiated by A. Newell et al. with the emergence of Artificial Intelligence. Significant advances in the logic deduction approach were made between 1950-60, e.g., in the work of H. Gelernter et al. Still, these advances are inefficient in proving relatively nontrivial theorems, such as those in classical geometry, where auxiliary lines are often necessary in the proofs. In the analytic approach, geometric theorems can be described by algebraic languages, leading to algebraic methods for automated theorem proving in geometry. Between 1970-80, W.-T. Wu proposed the Ritt-Wu characteristic set method for automated theorem proving, which was a breakthrough in Automated Reasoning at the time. Afterwards, several other algebraic methods for automated theorem proving were proposed, e.g., Gröbner basis method, area method, vector algebra method, Geometric Algebra method, to name a few. The scope of automatically provable geometric theorems also extends from classical Euclidean geometry and differential geometry to Riemannian geometry, non-Euclidean geometry, geometric inequalities, combinatorial identities, etc. From the viewpoint of knowledge management and query, the logic approach represents a geometric theorem as a logic implication of the conclusion by the hypotheses, both of which are relations among geometric objects. In this approach, applying the geometric theorem to a given geometric configuration is by first searching the database of geometric relations in the geometric configuration to match the hypotheses, then adding the conclusion to the database after exact match is fulfilled.