Journal of Automated Reasoning

, Volume 23, Issue 1, pp 63–82 | Cite as

Automatic Discovery of Theorems in Elementary Geometry

  • T. Recio
  • M. P. Vélez

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.

automatic theorem proving elementary geometry Gröbner basis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chou, S. C.: Mechanical Geometry Theorem Proving, D. Reidel, 1987.Google Scholar
  2. 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. 3.
    Conti, P. and Traverso, C.: A case of automatic theorem proving in Euclidean geometry: The Maclane 83 theorem, in Proceedings AAECC, Paris, 1995.Google Scholar
  4. 4.
    Cox, D., Little, J. and O'Shea, D.: Ideals, Varieties and Algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, 1992.Google Scholar
  5. 5.
    Guergueb, A.: Examples de demostration automatique en g è om é trie r é elle, Ph. Dissertation, Univ. de Rennes I, 1994.Google Scholar
  6. 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.Google Scholar
  7. 7.
    Kapur, D.: A refutational approach to theorem proving in geometry, Artif. Intell. 37(1–3) (1988), 61–93.Google Scholar
  8. 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. 9.
    Koepf, W.: Gröbner bases and triangles. Internat. J. Comput. Algebra in Math. Education 4(4) (1998), 371–386.Google Scholar
  10. 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.Google Scholar
  11. 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.Google Scholar
  12. 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.Google Scholar
  13. 13.
    Recio, T.: C á lculo Simb ó lico y Geom é trico, Editorial SÍntesis, Madrid, 1998.Google Scholar
  14. 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.Google Scholar
  15. 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. 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.Google Scholar
  17. 17.
    Wu, W. T.: Mechanical Theorem Proving in Geometries, Texts and Monographs in Symbolic Computation, Springer-Verlag, Wien, 1994.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • T. Recio
    • 1
  • M. P. Vélez
    • 2
  1. 1.Dep. de MatemáticasUniv. de CantabriaSantanderSpain
  2. 2.Dep. de Ingeniería InformáticaUniv. Antonio de NebrijaMadridSpain

Personalised recommendations