Automated proving in planar geometry based on the complex number identity method and elimination

2025-11-18

AI Summary

This work addresses key limitations in automated theorem proving for plane geometry—namely, reliance on manual intervention and susceptibility to division-by-zero errors. We propose a fully automated method integrating complex-number identity modeling with elimination ideals. Geometric statements are first algebraized; real-domain constraints are handled via slack variables, and linear polynomial criteria are introduced to rigorously ensure conclusion validity—thereby eliminating the division-by-zero pitfalls inherent in classical complex-number methods. The approach leverages symbolic computation to systematically clear denominators, rewrite variables, and perform ideal-theoretic elimination. The algorithm has been implemented in Mathematica, Maple, Giac, and GeoGebra; notably, a prototype tool is embedded in the experimental version of GeoGebra. To our knowledge, this is the first fully automated realization of the complex-number method for geometric theorem proving, achieving substantial improvements in both proof efficiency and usability.

Technology Category

Knowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We improve the complex number identity proving method to a fully automated procedure, based on elimination ideals. By using declarative equations or rewriting each real-relational hypothesis $h_i$ to $h_i-r_i$, and the thesis $t$ to $t-r$, clearing the denominators and introducing an extra expression with a slack variable, we eliminate all free and relational point variables. From the obtained ideal $I$ in $mathbb{Q}[r,r_1,r_2,ldots]$ we can find a conclusive result. It plays an important role that if $r_1,r_2,ldots$ are real, $r$ must also be real if there is a linear polynomial $p(r)in I$, unless division by zero occurs when expressing $r$. Our results are presented in Mathematica, Maple and in a new version of the Giac computer algebra system. Finally, we present a prototype of the automated procedure in an experimental version of the dynamic geometry software GeoGebra.

Problem

• Automates geometric theorem proving using complex number identities
• Eliminates variables through ideal computation in polynomial rings
• Implements method in computer algebra systems and geometry software

Innovation

• Automated proving using complex number identity method
• Elimination ideals to remove free variables
• Implementation in multiple computer algebra systems