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Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 78–94 | Cite as

Self-evident Automated Proving Based on Point Geometry from the Perspective of Wu’s Method Identity

  • Jingzhong Zhang
  • Xicheng Peng
  • Mao ChenEmail author
Article

Abstract

The algebraic methods represented by Wu’s method have made significant breakthroughs in the field of geometric theorem proving. Algebraic proofs usually involve large amounts of calculations, thus making it difficult to understand intuitively. However, if the authors look at Wu’s method from the perspective of identity,Wu’s method can be understood easily and can be used to generate new geometric propositions. To make geometric reasoning simpler, more expressive, and richer in geometric meaning, the authors establish a geometric algebraic system (point geometry built on nearly 20 basic properties/formulas about operations on points) while maintaining the advantages of the coordinate method, vector method, and particle geometry method and avoiding their disadvantages. Geometric relations in the propositions and conclusions of a geometric problem are expressed as identical equations of vector polynomials according to point geometry. Thereafter, a proof method that maintains the essence of Wu’s method is introduced to find the relationships between these equations. A test on more than 400 geometry statements shows that the proposed proof method, which is based on identical equations of vector polynomials, is simple and effective. Furthermore, when solving the original problem, this proof method can also help the authors recognize the relationship between the propositions of the problem and help the authors generate new geometric propositions.

Keywords

Geometry algebra point geometry proof method based on identical equations vector geometry Wu’s method 

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References

  1. [1]
    Wu W T, Mechanical Theorem Proving in Geometries: Basic Principles, Springer, New York, 1994.CrossRefGoogle Scholar
  2. [2]
    Wu W T, Mathematics Mechanization, Science Press, Kluwer, 2000.Google Scholar
  3. [3]
    Chou S C, Gao X S, and Zhang J Z, Machine Proofs in Geometry, World Scientific, Singapore, 1994.CrossRefzbMATHGoogle Scholar
  4. [4]
    Jiang J G and Zhang J Z, A review and prospect of readable machine proofs for geometry theorems, Journal of Systems Science & Complexity, 2012, 25(4): 802–820.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Leibniz, Math. Schriften, Berlin 1819, Vol. II: 17 & Vol. V: 133.Google Scholar
  6. [6]
    Grassmann, Geometrische Analyse, Nabu Press, Leipzig, 1847.Google Scholar
  7. [7]
    Li H B, Hestenes D, and Rockwood A, Generalized Homogeneous Coordinates for Computational Geometry, Ed. by Summer G, Geometric Computing with Clifford Algebras, Springer, Heidelberg, 2001.Google Scholar
  8. [8]
    Li H B, Invariant Algebras and Geometric Reasoning, World Scientific, Singapore, 2008.CrossRefzbMATHGoogle Scholar
  9. [9]
    Zhang N and Li H B, Affine bracket algebra theory and algorithms and their applications in mechanical theorem proving, Science in China (Series A: Mathematics), 2007, 50(7): 941–950.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Mo S K, Particle Geometry, Chongqing Press, Chongqing, 1992.Google Scholar
  11. [11]
    Wang K and Su Z, Automated geometry theorem proving for human-readable proofs, Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015), International Conference on Artificial Intelligence AAAI Press, 2015, 1193–1199.Google Scholar
  12. [12]
    Chen X Y, Song D, and Wang D M, Automated generation of geometric theorems from images of diagrams, Annals of Mathematics & Artificial Intelligence, 2015, 74(3–4): 333–358.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Ye Z, Chou S C, and Gao X S, Visually dynamic presentation of proofs in plane geometry, Journal of Automated Reasoning, 2010, 45(3): 213–241.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Yang L and Xia B C, Automated Proving and Discovering on Inequalities, Science Press, Beijing, 2008.Google Scholar
  15. [15]
    Chou S C, Gao X S, and Zhang J Z, Mechanical theorem proving by vector calculation, Proc. of International Symposium on Symbolic and Algebraic Computing, Kelantan, 1993, 284–291.Google Scholar
  16. [16]
    Zou Y and Zhang J Z, Automated generation of readable proofs for constructive geometry statements with the mass point method, Proc. of the 8th International Workshop on Automated Deduction in Geometry (ADG 2010), LNAI 6877, Springer-Verlag, Berlin Heidelberg, Germany, 2011, 221–258.CrossRefGoogle Scholar
  17. [17]
    Li T, Zou Y, and Zhang J Z, Improvement of the complex mass point method and its application in automated geometry theorem proving, Chinese Journal of Computers, 2015, 38(8): 1640–1647.MathSciNetGoogle Scholar
  18. [18]
    Ge Q, Zhang J Z, Chen M, et al., Automated geometry readable proving based on vector, Chinese Journal of Computers, 2014, 37(8): 1809–1819.MathSciNetGoogle Scholar
  19. [19]
    Zou Y, Fu Y H, and Zhang J Z, Preliminary study on the basis of geometric algebra from a new perspective, Journal of Systems Science and Mathematical Sciences, 2010, 30(1): 1–11.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Zhang J Z, Outlines for point-geometry, Studies in College Mathematics, 2018, 21(1): 1–8.Google Scholar
  21. [21]
    Jiang J G, Zhang J Z, and Wang X J, Readable proving for geometric theorems of polynomial equality type: Readable proving for geometric theorems of polynomial equality type, Chinese Journal of Computers, 2008, 31(2): 207–213.MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.National Engineering Research Center for E-LearningCentral China Normal UniversityWuhanChina
  2. 2.Institute of Computational Science and TechnologyGuangzhou UniversityGuangzhouChina
  3. 3.Chongqing Institute of Green and Intelligent TechnologyChinese Academy of SciencesChongqingChina

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