Skip to main content
SpringerLink
Log in

Automatic Discovery of Theorems in Elementary Geometry

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

We present here a further development of the well-known approach to automatic theorem proving in elementary geometry via algorithmic commutative algebra and algebraic geometry. Rather than confirming/refuting geometric statements (automatic proving) or finding geometric formulae holding among prescribed geometric magnitudes (automatic derivation), in this paper we consider (following Kapur and Mundy) the problem of dealing automatically with arbitrary geometric statements (i.e., theses that do not follow, in general, from the given hypotheses) aiming to find complementary hypotheses for the statements to become true. First we introduce some standard algebraic geometry notions in automatic proving, both for self-containment and in order to focus our own contribution. Then we present a rather successful but noncomplete method for automatic discovery that, roughly, proceeds adding the given conjectural thesis to the collection of hypotheses and then derives some special consequences from this new set of conditions. Several examples are discussed in detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chou, S. C.: Mechanical Geometry Theorem Proving, D. Reidel, 1987.

  2. Chou, S. S. and Gao, X. S.: Methods for mechanical geometry formula deriving, in Proceedings ISSAC-90, ACM, New York, 1990, pp. 265–270.

    Google Scholar 

  3. Conti, P. and Traverso, C.: A case of automatic theorem proving in Euclidean geometry: The Maclane 83 theorem, in Proceedings AAECC, Paris, 1995.

  4. Cox, D., Little, J. and O'Shea, D.: Ideals, Varieties and Algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, 1992.

  5. Guergueb, A.: Examples de demostration automatique en g è om é trie r é elle, Ph. Dissertation, Univ. de Rennes I, 1994.

  6. Kapur, D.: Geometry theorem proving using Hilbert's Nullstellensatz, in B.W. Char (ed.), Proc. of the 1986 Symposium on Symbolic and Algebraic Computation, Waterloo, Ont., 1986.

  7. Kapur, D.: A refutational approach to theorem proving in geometry, Artif. Intell. 37(1–3) (1988), 61–93.

    Google Scholar 

  8. Kapur, D. and Mundy, J. L.: Wu's method and its application to perspective viewing, in D. Kapur and J. L. Mundy (eds.), Geometric Reasoning, The MIT Press, Cambridge, Mass., 1989.

    Google Scholar 

  9. Koepf, W.: Gröbner bases and triangles. Internat. J. Comput. Algebra in Math. Education 4(4) (1998), 371–386.

    Google Scholar 

  10. Kutzler, B. and Stifter, S.: Automated geometry theorem proving using Buchberger's algorithm, in B. W. Char (ed.), Proc. of the 1986 Symposium on Symbolic and Algebraic Computation, Waterloo, Ont., 1986.

  11. Laborde, J. M. and Strässer, R.: Cabri Géomètre, a microworld of geometry for guided discovery learning, Zentralblatt f ü r Didaktik der Mathematik 90(5), 171–190.

  12. Mainguené, J.: M é thode de Wu, courbes r é elles et demostration automatique en G è om é trie, Ph. Dissertation, Univ. de Rennes I, 1994.

  13. Recio, T.: C á lculo Simb ó lico y Geom é trico, Editorial SÍntesis, Madrid, 1998.

    Google Scholar 

  14. Recio, T., Sterk, H. and Vélez, M. P.: Automatic geometry theorem proving, in A. M. Cohen, H. Cuipers, and H. Sterk (eds.), Some Tapas of Computer Algebra, Algorithms and Computations in Math. 4, Springer-Verlag, 1998.

  15. Wang, D.: Gröbner bases applied to geometric theorem proving and discovering, in B. Buchberger and F. Winkler (eds.), Gr ö bner Bases and Applications, London Math. Soc. Lecture Note Series, Cambridge University Press, Cambridge, 1998.

    Google Scholar 

  16. Wu, W. T.: On the decision problem and the mechanization of theorem-proving in elementary geometry, Scientia Sinica 21, 159–172 (also in W. W. Bledsoe and D. W. Loveland (eds.), Automated Theorem Proving: After 25 Years, American Mathematical Society, Providence, 1984, pp. 213–234.

  17. Wu, W. T.: Mechanical Theorem Proving in Geometries, Texts and Monographs in Symbolic Computation, Springer-Verlag, Wien, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dep. de Matemáticas, Univ. de Cantabria, 39071, Santander, Spain

    T. Recio

  2. Dep. de Ingeniería Informática, Univ. Antonio de Nebrija, 28040, Madrid, Spain

    M. P. Vélez

Authors
  1. T. Recio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. P. Vélez
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Recio, T., Vélez, M.P. Automatic Discovery of Theorems in Elementary Geometry. Journal of Automated Reasoning 23, 63–82 (1999). https://doi.org/10.1023/A:1006135322108

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1006135322108

Search

Navigation