Towards Understanding Triangle Construction Problems

  • Vesna Marinković
  • Predrag Janičić
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7362)


Straightedge and compass construction problems are one of the oldest and most challenging problems in elementary mathematics. The central challenge, for a human or for a computer program, in solving construction problems is a huge search space. In this paper we analyze one family of triangle construction problems, aiming at detecting a small core of the underlying geometry knowledge. The analysis leads to a small set of needed definitions, lemmas and primitive construction steps, and consequently, to a simple algorithm for automated solving of problems from this family. The same approach can be applied to other families of construction problems.


Triangle construction problems automated deduction in geometry 


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  1. 1.
    Adler, A.: Theorie der geometrischen konstruktionen, Göschen (1906)Google Scholar
  2. 2.
    Anglesio, J., Schindler, V.: Solution to problem 10719. American Mathematical Monthly 107, 952–954 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beeson, M.: Constructive geometry. In: Proceedings of the Tenth Asian Logic Colloquium. World Scientific (2010)Google Scholar
  4. 4.
    Berzsenyi, G.: Constructing triangles from three given parts. Quantum 396 (July/August 1994)Google Scholar
  5. 5.
    Connelly, H.: An extension of triangle constructions from located points. Forum Geometricorum 9, 109–112 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Connelly, H., Dergiades, N., Ehrmann, J.-P.: Construction of triangle from a vertex and the feet of two angle bisectors. Forum Geometricorum 7, 103–106 (2007)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Danneels, E.: A simple construction of a triangle from its centroid, incenter, and a vertex. Forum Geometricorum 5, 53–56 (2005)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Davis, P.J.: The rise, fall, and possible transfiguration of triangle geometry: A mini-history. The American Mathematical Monthly 102(3), 204–214 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    DeTemple, D.W.: Carlyle circles and the lemoine simplicity of polygon constructions. The American Mathematical Monthly 98(2), 97–108 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Djorić, M., Janičić, P.: Constructions, instructions, interactions. Teaching Mathematics and its Applications 23(2), 69–88 (2004)CrossRefGoogle Scholar
  11. 11.
    Fursenko, V.B.: Lexicographic account of triangle construction problems (part i). Mathematics in Schools 5, 4–30 (1937)Google Scholar
  12. 12.
    Fursenko, V.B.: Lexicographic account of triangle construction problems (part ii). Mathematics in schools 6, 21–45 (1937)Google Scholar
  13. 13.
    Gao, X.-S., Chou, S.-C.: Solving geometric constraint systems. I. A global propagation approach. Computer-Aided Design 30(1), 47–54 (1998)CrossRefGoogle Scholar
  14. 14.
    Gao, X.-S., Chou, S.-C.: Solving geometric constraint systems. II. A symbolic approach and decision of Rc-constructibility. Computer-Aided Design 30(2), 115–122 (1998)CrossRefGoogle Scholar
  15. 15.
    Grima, M., Pace, G.J.: An Embedded Geometrical Language in Haskell: Construction, Visualisation, Proof. In: Proceedings of Computer Science Annual Workshop (2007)Google Scholar
  16. 16.
    Gulwani, S., Korthikanti, V.A., Tiwari, A.: Synthesizing geometry constructions. In: Programming Language Design and Implementation, PLDI 2011, pp. 50–61. ACM (2011)Google Scholar
  17. 17.
    Chen, G.: Les Constructions Géométriques á la Régle et au Compas par une Méthode Algébrique. Master thesis, University of Strasbourg (1992)Google Scholar
  18. 18.
    Holland, G.: Computerunterstützung beim Lösen geometrischer Konstruktionsaufgaben. ZDM Zentralblatt für Didaktik der Mathematik 24(4) (1992)Google Scholar
  19. 19.
    Janičić, P.: GCLC — A Tool for Constructive Euclidean Geometry and More Than That. In: Iglesias, A., Takayama, N. (eds.) ICMS 2006. LNCS, vol. 4151, pp. 58–73. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Janičić, P., Quaresma, P.: System Description: GCLCprover + GeoThms. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 145–150. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Janičić, P.: Geometry Constructions Language. Journal of Automated Reasoning 44(1-2), 3–24 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Janičić, P., Narboux, J., Quaresma, P.: The area method: a recapitulation. Journal of Automated Reasoning 48(4), 489–532 (2012)CrossRefGoogle Scholar
  23. 23.
    Lebesgue, H.-L.: Leçons sur les constructions géométriques. Gauthier-Villars (1950)Google Scholar
  24. 24.
    Lopes, L.: Manuel de Construction de Triangles. QED Texte (1996)Google Scholar
  25. 25.
    Marić, F., Petrović, I., Petrović, D., Janičić, P.: Formalization and implementation of algebraic methods in geometry. Electronic Proceedings in Theoretical Computer Science 79 (2012)Google Scholar
  26. 26.
    Martin, G.E.: Geometric Constructions. Springer (1998)Google Scholar
  27. 27.
    Meyers, L.F.: Update on William Wernick’s “triangle constructions with three located points”. Mathematics Magazine 69(1), 46–49 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pambuccian, V.: Axiomatizing geometric constructions. Journal of Applied Logic 6(1), 24–46 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schreck, P.: Constructions à la règle et au compas. PhD thesis, University of Strasbourg (1993)Google Scholar
  30. 30.
    Specht, E.: Wernicks liste (in German),
  31. 31.
    Stewart, I.: Galois Theory. Chapman and Hall Ltd. (1973)Google Scholar
  32. 32.
    Stojanović, S., Pavlović, V., Janičić, P.: A Coherent Logic Based Geometry Theorem Prover Capable of Producing Formal and Readable Proofs. In: Schreck, P., Narboux, J., Richter-Gebert, J. (eds.) ADG 2010. LNCS, vol. 6877, pp. 201–220. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  33. 33.
    Ustinov, A.V.: On the construction of a triangle from the feet of its angle bisectors. Forum Geometricorum 9, 279–280 (2009)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Wernick, W.: Triangle constructions vith three located points. Mathematics Magazine 55(4), 227–230 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yiu, P.: Elegant geometric constructions. Forum Geometricorum 5, 75–96 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Vesna Marinković
    • 1
  • Predrag Janičić
    • 1
  1. 1.Faculty of MathematicsUniversity of BelgradeSerbia

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