Towards Solving the Dynamic Geometry Bottleneck Via a Symbolic Approach

  • Francisco Botana
  • Tomás Recio
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3763)


The goal of this paper is to report on a prototype of a new dynamic geometry software, GDI (Geometría Dinámica Inteligente). We will describe how, apart from being a standard dynamic environment for elementary geometry, GDI addresses some key problems of the dynamic geometry paradigm, by including enhanced tools for loci generation and automatic proving, plus another distinguished feature, namely, a discovery option, allowing the user to find complementary hypotheses for arbitrary statements to become true. The key technique for all these improvements is the development of an automatic “bridge” between the graphic and the algebraic counterparts of the program (calling on an external computer algebra system).


Elementary Geometry Dynamic Geometry Automate Deduction Dynamic Geometry Software Symbolic Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bazzotti, L., Dalzotto, G., Robbiano, L.: Remarks on geometric theorem proving. In: Richter-Gebert, J., Wang, D. (eds.) ADG 2000. LNCS (LNAI), vol. 2061, pp. 104–128. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Botana, F.: Interactive versus symbolic approaches to plane loci generation in dynamic geometry environments. In: Sloot, P.M.A., Tan, C.J.K., Dongarra, J., Hoekstra, A.G. (eds.) ICCS-ComputSci 2002. LNCS, vol. 2330, pp. 211–218. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Botana, F., Valcarce, J.L.: A software tool for the investigation of plane loci. Mathematics and Computers in Simulation 61(2), 141–154 (2003)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Botana, F., Valcarce, J.L.: Automatic determination of envelopes and other derived curves within a graphic environment. Mathematics and Computers in Simulation 67(1–2), 3–13 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Capani, A., Niesi, G., Robbiano, L.: CoCoA, a system for doing Computations in Commutative Algebra. Available via anonymous ftp from,
  6. 6.
    Conti, P., Traverso, C.: Algebraic and semialgebraic proofs: Methods and paradoxes. In: Richter-Gebert, J., Wang, D. (eds.) ADG 2000. LNCS (LNAI), vol. 2061, pp. 83–103. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Dolzmann, A., Sturm, A., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. Journal of Automated Reasoning 21, 357–380 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gawlick, T.: Towards a theory of visualization by dynamic geometry software. Paradigms, phenomena, principles. Int. Conf. on Mathematical Education 10 (ICME 10). Topic Study Group 16,
  9. 9.
    Gao, X.S., Chou, S.C.: Solving geometric constraint systems. I. A global propagation approach. Computer–Aided Design 30, 47–54 (1998)CrossRefGoogle Scholar
  10. 10.
    Gao, X.S., Chou, S.C.: Solving geometric constraint systems. II. A symbolic approach and decision of Rc–constructibility. Computer–Aided Design 30, 115–122 (1998)CrossRefGoogle Scholar
  11. 11.
    Gao, X.S., Zang, J.Z., Chou, S.C.: Geometry expert. Nine Chapters Publ., Taiwan (1998)Google Scholar
  12. 12.
    Giering, O.: Affine and projective generalization of Wallace lines. Journal of Geometry and Graphics 1(2), 119–133 (1997)zbMATHMathSciNetGoogle Scholar
  13. 13.
    González López, M.J.: Using dynamic geometry software to simulate physical motion. International Journal of Computers for Mathematical Learning 6(2), 127–142 (2001)CrossRefGoogle Scholar
  14. 14.
    de Guzmán, M.: An extension of the Wallace–Simson theorem: projecting in arbitrary directions. American Mathematical Monthly 106(6), 574–580 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hoffmann, C., Bouma, W., Fudos, I., Cai, J., Paige, R.: A geometric constraint solver. Computer–Aided Design 27, 487–501 (1995)zbMATHGoogle Scholar
  16. 16.
    Jackiw, N.: The Geometer’s Sketchpad. Key Curriculum Press, Berkeley (1997)Google Scholar
  17. 17.
    King, J., Schattschneider, D.: Geometry turned on Mathematical Association of America, Washington DC (1997)Google Scholar
  18. 18.
    Kortenkamp, U.: Foundations of dynamic geometry, Ph.D. Thesis, ETH, Zurich (1999)Google Scholar
  19. 19.
    Kortenkamp, U., Richter-Gebert, J.: A dynamic setup for elementary geometry. In: Proc. Multimedia Tools for Comunicating Mathematics (MTCM 2000). Springer, Heidelberg (2001)Google Scholar
  20. 20.
    Laborde, J.M., Bellemain, F.: Cabri Geometry II. Texas Instruments, Dallas (1998)Google Scholar
  21. 21.
    Laborde, J.M.: Some issues raised by the development of implemented dynamic geometry as with Cabri-Geometre. In: Proc. 15th European Workshop on Computational Geometry, pp. 7–19 (1999)Google Scholar
  22. 22.
    MacLane, S.: Some interpretation of abstract linear dependence in terms of projective geometry. American Journal of Mathematics 58, 236–240 (1936)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Recio, T.: Cálculo simbólico y geométrico. Síntesis, Madrid (1998)Google Scholar
  24. 24.
    Recio, T., Botana, F.: Where the truth lies (in automatic geometry theorem proving). In: Laganá, A., Gavrilova, M.L., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds.) ICCSA 2004. LNCS, vol. 3044, pp. 761–770. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  25. 25.
    Recio, T., Sterk, H., Vélez, P.: Project: Automated Geometry theorem proving. In: Cohen, A.M., Cuypers, H., Sterk, H. (eds.) Some Tapas of Computer Algebra (Algorithms and Computations in Mathematics, vol. 4, pp. 276–296. Springer, Berlin (1999)Google Scholar
  26. 26.
    Recio, T., Vélez, M.P.: Automatic discovery of theorems in elementary geometry. Journal of Automated Reasoning 23, 63–82 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Richard, P.: Raisonnement et stratégies de preuve dans l’enseignement des mathématiques, Peter Lang, Berne (2004)Google Scholar
  28. 28.
    Richter–Gebert, J., Kortenkamp, U.: The interactive geometry software Cinderella. Springer, Berlin (1999)Google Scholar
  29. 29.
    Wang, D.: GEOTHER: A geometry theorem prover. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 166–170. Springer, Heidelberg (1996)Google Scholar
  30. 30.
    Wang, D., Zhi, L.: Algebraic Factorization Applied to Geometric Problems. In: Proc. Third Asian Symposium on Computer Mathematics (ASCM 1998), pp. 23–36. Lanzhou University Press, Lanzhou (1998)Google Scholar
  31. 31.

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Francisco Botana
    • 1
  • Tomás Recio
    • 2
  1. 1.Departamento de Matemática Aplicada IUniversidad de VigoPontevedraSpain
  2. 2.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain

Personalised recommendations