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Is Computer Algebra Ready for Conjecturing and Proving Geometric Inequalities in the Classroom?

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Abstract

We introduce an experimental version of GeoGebra that successfully conjectures and proves a large scale of geometric inequalities by providing an easy-to-use graphical interface. GeoGebra Discovery includes an embedded version of the Tarski/QEPCAD B system which can solve a real quantifier elimination problem, so the input geometric construction can be translated into a semi-algebraic system, and after some algebraic manipulations, the obtained formula can be translated back to a precisely stated geometric inequality. We provide some non-trivial examples to illustrate the performance of GeoGebra Discovery when dealing with inequalities, as well as some technical difficulties.

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Notes

  1. See [2]. It is inequality 4.2. on page 31. We have followed the precise formulation of this author, a well known authority in the field of triangle inequalities.

  2. Available for download at https://www.cypress.io/.

  3. See https://matek.hu/zoltan/compare-20211228/all.html for the detailed results.

  4. See https://matek.hu/zoltan/proverrg-20211228/all.html.

References

  1. Abar, C. A. A. P., Kovács, Z., Recio, T., Vajda, R.: Conectando Mathematica e GeoGebra para explorar construções geométricas planas. Presentation at Wolfram Technology Conference, Saõ Paulo, Brazil, November (2019)

  2. Bottema, O.: Inequalities for \(R\), \(r\) and \(s\). Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika, 338/352:27–36, (1971)

  3. Bottema, O., Djordjevic, R., Janic, R., Mitrinovic, D., Vasic, P.: Geometric Inequalities. Wolters-Noordhoff Publishing, Groningen (1969)

    MATH  Google Scholar 

  4. Bright, P.: The Web is getting its bytecode: WebAssembly. Condé Nast (2015)

  5. Brown, C., Davenport, J.: The complexity of quantifier elimination and cylindrical algebraic decomposition. In Proceedings of ISSAC ’07, 54–60. ACM, (2007)

  6. Brown, C.W.: An overview of QEPCAD B: a tool for real quantifier elimination and formula simplification. J. Jpn. Soc. Symb. Algebr. Comput. 10(1), 13–22 (2003)

    Google Scholar 

  7. Bulmer, M., Fearnley-Sander, D., Stokes, T.: The kinds of truth of geometry theorems. In: Richter-Gebert, J., Wang, D. (eds.) Automated Deduction in Geometry. Lecture Notes in Computer Science 2061, pp. 129–143. Springer, Berlin (2001)

    MATH  Google Scholar 

  8. Chen, C., Maza, M.M.: Quantifier elimination by cylindrical algebraic decomposition based on regular chains. J. Symb. Comput. 75, 74–93 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chou, S.C.: Mechanical Geometry Theorem Proving. D. Reidel Publishing Company, Dordrecht, Netherlands (1988)

    MATH  Google Scholar 

  10. Collins, G.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. Lect. Notes Comput. Sci. 33, 134–183 (1975)

    Article  Google Scholar 

  11. Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12, 299–328 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Conti, P., Traverso, C.: A case of automatic theorem proving in Euclidean geometry: the Maclane 83 theorem. In: Cohen, G., Giusti, M., Mora, T. (eds.) Applied Algebra. Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, Springer, Berlin (1995)

    Google Scholar 

  13. Conti, P., Traverso, C.: Algebraic and semialgebraic proofs: methods and paradoxes. In: Richter-Gebert, J., Wang, D. (eds.) Automated Deduction in Geometry. Lecture Notes in Computer Science, pp. 83–103. Springer, Berlin (2001)

    Chapter  Google Scholar 

  14. Davenport, J., Heintz, J.: Real quantifier elimination is doubly exponential. J. Symb. Comput., 5(1)

  15. Dolzmann, A., Sturm, T.: Redlog: computer algebra meets computer logic. ACM SIGSAM Bull. 31(2), 2–9 (1997)

    Article  Google Scholar 

  16. Dolzmann, A., Sturm, T., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. J. Autom. Reason. 21(3), 357–380 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Erdős, P.: Problem 3740. Amer. Math. Mon. 42, 396 (1935)

    Google Scholar 

  18. Guergueb, A., Mainguené, J., Roy, M.F.: Examples of automatic theorem proving in real geometry. In: ISSAC-94, 20–23, ACM press, (1994)

  19. Hanna, G., Yan, X.: Opening a discussion on teaching proof with automated theorem provers. Learn. Math. 41(3), 42–46 (2021)

    Google Scholar 

  20. Hohenwarter, M., Kovács, Z., Recio, T.: Using Automated Reasoning Tools to Explore Geometric Statements and Conjectures, 215–236. Springer International Publishing, Cham (2019)

    Google Scholar 

  21. Iwane, H., Yanami, H., Anai, H.: SyNRAC: a toolbox for solving real algebraic constraints. In Proceedings of ICMS-2014. LNCS, vol. 8592

  22. Kovács, Z.: GeoGebra Discovery. A GitHub project, 07 2020. https://github.com/kovzol/geogebra-discovery

  23. Kovács, Z., Recio, T., Richard, P. R., Vaerenbergh, S. V., Vélez, M. P.: Towards an ecosystem for computer-supported geometric reasoning. Int. J. Math. Edu. Sci. Technol. (2020)

  24. Kovács, Z., Recio, T., Tabera, L.F., Vélez, M.P.: Dealing with degeneracies in automated theorem proving in geometry. Mathematics 9(16), 1964 (2021)

    Article  Google Scholar 

  25. Kovács, Z., Recio, T., Vélez, M.P.: Using automated reasoning tools in GeoGebra in the teaching and learning of proving in geometry. Int. J. Technol. Math. Edu. 25(2), 33–50 (2018)

    Google Scholar 

  26. Kovács, Z., Recio, T., Vélez, M.P.: Detecting truth, just on parts. Revista Matemática Complutense 32(2), 451–474 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kovács, Z., Recio, T., Vélez, M. P.: Approaching Cesáro’s inequality through GeoGebra Discovery”. In: Proceedings of the 26th Asian Technology Conference in Mathematics, W. C. Yang, D. B. Meade, M. Majewski (eds), 160–174. Mathematics and Technology, LL, Dec. 13–15, 2021

  28. M. Ladra, P. Páez-Guillán, T. Recio. Dealing with negative conditions in automated proving: tools and challenges. The unexpected consequences of Rabinowitsch’s trick. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales (RACSAM), 114(4), 2020

  29. Mordell, L.J., Barrow, D.F.: Solution to 3740. Amer. Math. Mon. 44, 252–254 (1937)

    Google Scholar 

  30. T. Recio, F. Botana. Where the truth lies (in Automatic Theorem Proving in Elementary Geometry). In: Proceedings of international conference on computational science and its applications 2004, Lecture Notes in Computer Science 3044:761–771, Springer, 2004

  31. Recio, T., Losada, R., Kovács, Z., Ueno, C.: Discovering geometric inequalities: the concourse of geoGebra discovery, dynamic coloring and maple tools. Mathematics 9(20), 2548 (2021)

    Article  Google Scholar 

  32. Recio, T., Vélez, M.P.: Automatic discovery of theorems in elementary geometry. J. Autom. Reason. 23, 63–82 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. Reiman, I.: Fejezetek az elemi geometriából. Typotex, Budapest (2002)

    Google Scholar 

  34. Sturm, T.: A survey of some methods for real quantifier elimination, decision, and satisfiability and their applications. Math. Comput. Sci. 11, 483–502 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tarski, A.A., Decision Method, A., for Elementary Algebra and Geometry. Manuscript. Santa Monica, CA: RAND Corp.,: Republished as A Decision Method for Elementary Algebra and Geometry, 2nd edn., p. 1951. University of California Press, Berkeley, CA (1948)

  36. Vajda, R., Kovács, Z.: GeoGebra and the realgeom reasoning tool. CEUR Workshop Proceedings 2752, 204–219 (2020)

  37. Vale-Enriquez, F., Brown, C.: Polynomial constraints and unsat cores in Tarski. In: Mathematical Software - ICMS 2018. LNCS, vol. 10931, pp. 466–474. Springer, Cham (2018)

  38. Wolfram Research, Inc. Mathematica, version 12.1, 2020. Champaign, IL

  39. Wu, W.T.: On the decision problem and the mechanization of theorem proving in elementary geometry. Sci. Sinica 21, 157–179 (1978)

    MathSciNet  Google Scholar 

  40. Xia, B., Yang, L.: Automated Inequality Proving And Discovering. World Scientific, Singapore (2017)

    MATH  Google Scholar 

  41. Yang, L., Xia, B.: Automated deduction in real geometry. In: Chen, F., Wang, D. (eds.) Lecture Notes Series on Computing Volume 11: Geometric Computation, pp. 248–298. World Scientific, Singapore (2004)

    Google Scholar 

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Acknowledgements

Second, third and fifth authors were partially supported by a grant PID2020-113192GB-I00 (Mathematical Visualization: Foundations, Algorithms and Applications) from the Spanish MICINN.

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Authors and Affiliations

  1. United States Naval Academy, 121 Blake Road, Annapolis, MD, 21402, USA

    Christopher W. Brown

  2. The Private University College of Education of the Diocese of Linz, Salesianumweg 3, 4020, Linz, Austria

    Zoltán Kovács

  3. Universidad Antonio de Nebrija, C/ Santa Cruz de Marcenado 27, 28015, Madrid, Spain

    Tomás Recio & M. Pilar Vélez

  4. Bolyai Institute, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    Róbert Vajda

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  1. Christopher W. Brown
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  2. Zoltán Kovács
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  3. Tomás Recio
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  4. Róbert Vajda
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  5. M. Pilar Vélez
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Correspondence to Tomás Recio.

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Brown, C.W., Kovács, Z., Recio, T. et al. Is Computer Algebra Ready for Conjecturing and Proving Geometric Inequalities in the Classroom?. Math.Comput.Sci. 16, 31 (2022). https://doi.org/10.1007/s11786-022-00532-9

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