Abstract
Automated discovery of geometric theorems has attracted considerable attention from the research community. In this paper, a new method is proposed to discover geometric theorems automatically. This method first generates vector equations based on given geometric relations about a geometric figure and then transforms the vector equations into a system of homogeneous linear equations; after computing the determinants of the coefficient matrices corresponding to the system of equations, the elimination method is applied to obtain a large number of geometric relationships. The test on more than 200 geometric problems shows that the geometric relationships discovered automatically by the proposed method are of obvious geometric meaning.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Jiang, J.G., Zhang, J.Z.: A review and prospect of readable machine proofs for geometry theorems. J. Syst. Sci. Complex. 25(4), 802–820 (2012)
Wu, W.T.: Mechanical Theorem Proving in Geometries: Basic Principles. Springer, New York (1994)
Wu, W.T.: Mathematics Mechanization. Science Press, Kluwer (2000)
Chou, S.C., Gao, X.S., Zhang, J.Z.: Machine Proofs in Geometry. World Scientific, Singapore (1994)
Nevins, A.J.: Plane geometry theorem proving using forward chaining. Artif. Intell. 6(1), 1–23 (1975)
Chou, S.C., Gao, X.S., Zhang, J.Z.: A deductive database approach to automated geometry theorem proving and discovering. J. Autom. Reason. 25(3), 219–246 (2000)
Li, S.Z., Wang, D.M., Zhang, J.Z.: Symbolic Computation and Education. World Scientific, Singapore (2007)
Botana, F., Hohenwarter, M., Janicic, P., Kovács, Z., Petrović, I., Recio, T., Weitzhofer, S.: Automated theorem proving in GeoGebra: current achievements. J. Autom. Reason. 55(1), 39–59 (2015)
Ye, Z., Chou, S.C., Gao, X.S.: Visually dynamic presentation of proofs in plane geometry. J. Autom. Reason. 45(3), 213–241 (2010)
Todd, P.: Geometry expressions: a constraint based interactive symbolic geometry system. In: Proc. of the 6th International Workshop on Automated Deduction in Geometry (ADG 2006). LNAI 6877. Springer, Berlin, Heidelberg, New York, pp. 189–202 (2007)
Botana, F., Valcarce, J.L.: A dynamic-symbolic interface for geometric theorem discovery. Comput. Educ. 38(1–3), 21–35 (2002)
Zhang, J.Z., Peng, X.C., Chen, M.: Self-evident automated proving based on point geometry from the perspective of Wu’s method identity. J. Syst. Sci. Complex. 32(1), 78–94 (2019)
Jie, Z., Wang, D., Yao, S.: Automated reducible geometric theorem proving and discovery by Gröbner basis method. J. Autom. Reason. 59(3), 1–14 (2016)
Chou, S.C.: Proving and Discovering Theorems in Elementary Geometries Using Wu’s Method Ph.D. Thesis, Department of Mathematics, University of Texas, Austin (1985)
Pech, P.: Selected Topics in Geometry with Classical vs. Computer Proving. World Scientific, Singapore (2007)
Recio, T., Vélez, M.P.: Automated discovering of theorems in elementary geometry. J. Autom. Reason. 23(1), 63–82 (1999)
Kutzler, B., Stifter, S.: On the application of Buchberger’s algorithm to automated geometry theorem proving. J. Symb. Comput. 2, 389–397 (1986)
Buchberger, B., Collins, G., Kutzler, B.: Algebraic methods for geometric reasoning. Annu. Rev. Comput. Sci. 3(19), 85–119 (1995)
Wang, D.M.: Elimination Methods. Springer, Wien New York (2001)
Wang, D.M.: Elimination Practice. Software Tools and Applications. Imperial College Press, London (2004)
Kapur, D., Saxena, T., Yang, L.: Algebraic and geometric reasoning with dixon resultants. In: Proceedings of International Symposium on Symbolic and Algebraic Computation (Oxford, 1994). ACM Press, New York, pp. 99–107 (1994)
Zhang, J.Z., Yang, L., Hou, X.R.: The sub-resultant method for automated theorem proving. J. Syst. Sci. Math. Sci. 15(1), 10–15 (1995)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms, Undergraduate Texts in Mathematics. Springer, Berlin (1992)
Kapur, D.: Automated geometric reasoning: Dixon resultants, Gröbner bases, and characteristic sets. In: Automated Deduction in Geometry. LNAI 1360. Springer, pp. 1–36 (1997)
Kapur, D.: Using Grobner bases to reason about geometry problems. J. Symb. Comput. 2(4), 399–408 (1986)
Kapur, D.: Geometry theorem proving using Hilbert's Nullstellensatz. In: Proc. of SYMSAC’86, Waterloo, pp. 202–208 (1986)
Johnson, R.A.: Advanced Euclidean Geometry. Houghton Mifflin Company, Boston (1929)
Apollonius: Conics, Books I–III. Green Lion Press, Santa Fe (1997)
Li, H.: Vectorial equations solving for mechanical geometry theorem proving. J. Autom. Reason. 25(2), 83–121 (2000)
Chou, S.C., Gao, X.S., Zhang, J.Z.: Mechanical geometry theorem proving by vector calculation. In: Proc. ISSAC93, Kiev. ACM Press, pp. 284–291 (1993)
Ge, Q., Zhang, J.Z., Chen, M., Peng, X.C.: Automated geometry readable proving based on vector. Chin. J. Comput. 37(8), 1809–1819 (2014)
Acknowledgements
This work was financially supported by the National Natural Science Foundation of China (Grant No. 62077019, 41671377), National Key R&D Program of China (Grant No. 2017YFB1401300). We are grateful to the reviewers for their useful comments and suggestions which helped us to significantly improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
Rights and permissions
About this article
Cite this article
Peng, X., Chen, Q., Zhang, J. et al. Automated Discovery of Geometric Theorems Based on Vector Equations. J Autom Reasoning 65, 711–726 (2021). https://doi.org/10.1007/s10817-021-09591-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10817-021-09591-2