Abstract
We present a new method for the systematic and automated finding of theorems holding on a given elementary geometry figure. The process is illustrated by means of the software tool Automated Geometer, developed by the authors on top of GeoGebra, a dynamic geometry system with millions of users at high schools and universities, all over the world, thus conferring our proposal with a potential impact beyond the scientific community context. The Automated Geometer exploits GeoGebra’s recently added new functionalities concerning the automated verification or denial of geometric properties conjectured by the user. We emphasize that the method for mechanically finding a complete list of geometric properties that hold on a user-provided construction, is purely symbolic, thus giving such properties rigorous mathematical certainty.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dalzotto, G., Recio, T.: On protocols for the automated discovery of theorems in elementary geometry. J. Autom. Reason. 43, 203–236 (2009)
Pech, P.: Selected Topics in Geometry with Classical Vs. World Scientific. Computer Proving (2007)
Botana, F., Kovács, Z., Recio, T.: Automated geometer, a web-based discovery tool. In: Hongbo Li (ed.), Proceedings of the 12th International Conference on Automated Deduction in Geometry (ADG) (2018) 7–13. Available at http://adg2018.cc4cm.org/ADG2018Proceedings
Botana, F., Kovács, Z., Recio, T.: Towards an Automated Geometer. Lect. Notes Artif. Intell. 11110, 215–220 (2018)
Lenat, D.B.: Automated theory formation in mathematics. Contemp. Math. 29, 287–314 (1984)
Zeilberger, D.: Plane Geometry: an elementary textbook by Shalosh B. Ekhad XIV (Circa 20150), downloaded from the future by Doron Zeilberger. http://sites.math.rutgers.edu/~zeilberg/PG/gt.html accessed 15/10/2018
de Guzmán, M.: La experiencia de descubrir en geometría. Nivola (2002)
Bagai, R., Shanbhogue, V., Zytkow, J.M., Chou, S.C.: Automatic theorem generation in plane geometry. Lect. Notes Artif. Intell. 689, 415–424 (1993)
Chou, S.C., Gao, X.S., Zhang, J.Z.: A deductive database approach to automated geometry theorem proving and discovering. J. Autom. Reason. 25, 219–246 (2000)
Chen, X., Song, D., Wang, D.: Automated generation of geometric theorems from images of diagrams. Ann. Math. Artif. Intell. 74, 333–358 (2015)
Song, D., Wang, D., Chen, X.: Retrieving geometric information from images: the case of hand-drawn diagrams. Data Min. Knowl. Disc. 31, 934 (2017). https://doi.org/10.1007/s10618-017-0494-1
Botana, F., Hohenwarter, M., Janičić, P., Kovács, Z., Petrović, I., Recio, T., Weitzhofer, S.: Automated theorem proving in GeoGebra: current achievements. J. Autom. Reason. 55, 39–59 (2015)
Kovács, Z., Recio, T., Vélez, M.P.: Detecting truth, just on parts. Revista Matemática Complutense 32, 451–474 (2019)
Magajna, Z.: OK Geometry. http://z-maga.si/index?action=article&id=40. Accessed 5 Mar 2018
Recio, T., Vélez, M.P.: Automatic discovery of theorems in elementary geometry. J. Autom. Reas. 23, 63–82 (1999)
The GeoGebra Team: Reference: GeoGebra Apps API. https://wiki.geogebra.org/en/Reference:GeoGebra_Apps_API. Accessed 5 Oct 2018
Kovács, Z., Parisse, B.: Giac and GeoGebra - Improved Gröbner basis computations. Lect. Notes Comput. Sci. 8942, 126–138 (2015)
Bright, P.: The Web is getting its bytecode: WebAssembly. Condé Nast (2015)
Brianchon, C. J., Poncelet J.-V.: Recherche sur la détermination d’une hyperbole équilatère au moyen de quatre conditions données, Annales de Gergonne 11 (1820-1821) 205-220. http://www.numdam.org/article/AMPA_1820-1821__11__205_0.pdf. Accessed 10 Sept 2018
Gao, H., Goto, Y., Cheng, J.: A systematic methodology for automated theorem finding. Theoret. Comput. Sci. 554, 2–21 (2014)
Gao, H., Li, J., Cheng, J.: Measuring interestingness of theorems in automated theorem finding by forward reasoning based on strong relevant logic. In: Proceedings 2019 IEEE International Conference on Energy Internet (ICEI), Nanjing, China, 2019, pp. 356–361
Gao, H., Goto, Y., Cheng, J.: A set of metrics for measuring interestingness of theorems in automated theorem finding by forward reasoning: A case study in NBG set theory. Proceedings of the International Conference on Intelligence Science and Big Data Engineering (2015). Part II. Lecture Notes in Computer Science 9243 (2015) 508–517
Botana, F., Kovács, Z., Martínez-Sevilla, A., Recio, T.: Automatically augmented reality with GeoGebra. In: Promodrou, T. (ed.) Augmented Reality in Educational Settings, pp. 347–368. Brill Sense, Leiden (2020)
Su, W., Cai, C., Wu, J.: The accessibility of mathematical formulas for the visually impaired in China. Lect. Notes Artif. Intell. 11110, 237–242 (2018)
Abánades, M., Botana, F., Kovács, Z., Recio, T., Sólyom-Gecse, C.: Development of automatic reasoning tools in GeoGebra. ACM Commun. Comput. Algebra 50, 85–88 (2016)
Acknowledgements
Authors partially supported by the Grant MTM2017-88796-P from the Spanish MINECO and the ERDF (European Regional Development Fund).
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.
Rights and permissions
About this article
Cite this article
Botana, F., Kovács, Z. & Recio, T. A Mechanical Geometer. Math.Comput.Sci. 15, 631–641 (2021). https://doi.org/10.1007/s11786-020-00497-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-020-00497-7