Skip to main content
Log in

Towards a Certified Version of the Encyclopedia of Triangle Centers

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

Triangle centers such as the center of gravity, the circumcenter, the orthocenter are well studied by geometers. Recently, under the guidance of Clark Kimberling, an electronic encyclopedia of triangle centers (ETC) has been developed, it contains more than 7000 centers and many properties of these points. In this paper, we describe how we created a certified version of ETC such that some of the properties described come along with a computer checked proof of its validity.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development, Coq’Art: The Calculus of Inductive Constructions. Springer, Texts in Theoretical Computer Science. An EATCS Series (2004)

    Book  MATH  Google Scholar 

  2. Braun, G., Narboux, J.: From Tarski to Hilbert. In: Ida, T., Fleuriot, J. (eds.) Post-proceedings of Automated Deduction in Geometry 2012 volume 7993 of LNCS, pp. 89–109. Springer, Edinburgh (2012)

    Google Scholar 

  3. Boutry, P., Narboux, J., Schreck, P., Braun, G.: Automated Deduction in Geometry 2014. Proceedings of ADG 2014. In: Botana, Francisco, Quaresma, Pedro (eds.) Using small scale automation to improve both accessibility and readability of formal proofs in geometry. pp 1–19, Coimbra, Portugal (2014)

  4. Chou, S.C., Gao, X.S.: Ritt-Wu’s Decomposition Algorithm and Geometry Theorem Proving. In: Stickel, Mark E. (eds.) CADE, volume 449 of Lecture Notes in Computer Science. pp 207–220, Springer (1990)

  5. Delahaye, D., Mayero, M.: Field, une procédure de décision pour les nombres réels en coq. In: JFLA, pp. 33–48 (2001)

  6. Gallatly, W.: The modern geometry of the triangle. In: Hodgson, F (ed) London (1910)

  7. Grégoire, B., Mahboubi, A.: Proving equalities in a commutative ring done right in coq. In: Hurd, J., Melham, Thomas F.: (eds), Theorem Proving in Higher Order Logics, 18th International Conference, TPHOLs 2005, Oxford, UK, August 22–25, 2005, Proceedings, volume 3603 of Lecture Notes in Computer Science. Springer, pp. 98–113 (2005)

  8. Kapur, D.: Geometry Theorem Proving using Hilbert’s Nullstellensatz. In: SYMSAC ’86: Proceedings of the fifth ACM symposium on Symbolic and algebraic computation. pp 202–208, ACM Press, New York (1986)

  9. Kimberling, C.: Triangle centers as functions. J. Math. 23(4) (1993)

  10. Kimberling, C.: Central points and central lines in the plane of a triangle. Math Mag, pp 163–187 (1994)

  11. Kimberling, C.: Triangle centers and central triangles (1998)

  12. Narboux, J.: Mechanical Theorem Proving in Tarski’s geometry. In: Eugenio, Francisco Botana, Lozano, Roanes: (eds.) Post-proceedings of Automated Deduction in Geometry 2006, vol 4869 of LNCS, Pontevedra, Spain, Francisco Botana, pp 139–156. Springer (2007)

  13. Rybowicz, M.: On the normalization of numbers and functions defined by radicals. J. Symbol. Comput. 35(6), 651–672 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, D.: Elimination procedures for mechanical theorem proving in geometry. Ann. Math. Artif. Intell. 13(1–2), 1–24 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yiu, P.: Introduction to the geometry of triangle (2002)

  16. Zippel, R.: Simplification of expressions involving radicals. J. Symbol. Comput. 1(2), 189–210 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Julien Narboux.

Additional information

The website is available here: http://dpt-info.u-strasbg.fr/~narboux/CETC/about.html.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Narboux, J., Braun, D. Towards a Certified Version of the Encyclopedia of Triangle Centers. Math.Comput.Sci. 10, 57–73 (2016). https://doi.org/10.1007/s11786-016-0254-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-016-0254-4

Keywords

Mathematics Subject Classification

Navigation