Abstract
With dynamic mediums such as computer displays, we propose a new kind of visually dynamic presentation of proofs in plane geometry. In a single diagram for the proof, when the proof text goes on step by step with mouse clicks, the related geometry elements in the diagram are added, animated, or deleted dynamically with various visually dynamic effects. It solves not only the problem of identifying geometry elements in the proof text with those in the diagram, but also makes the proof more vividly visualized and intuitive. Our ongoing developing system “Java Geometry Expert” (JGEX) uses two methods to create such visually dynamic presentations: the manual input method and the automatic method. In this first part of the series of our work, we propose the main features of our visually dynamic presentation of proofs and present the manual input method to create such presentations. The manual input method mainly uses mouse clicks to create the dynamic geometry diagram and the proof text.
Similar content being viewed by others
References
Nelsen, R.B.: Proofs without Words: Exercises in Visual Thinking (Classroom Resource Material) (1993)
Nelsen, R.B.: Proofs without Words II: more Exercises in Visual Thinking (Classroom Resource Material), by Roger B. Nelsen (2001)
Alsina, C., Nelsen, R.B.: [C] Math Made Visual: Creating Images For Understanding Mathematics. The Mathematical Association of America, New York (2006)
Cut-The-Knot: Geometry articles, theorems, problems. http://www.cut-the-knot.org/geometry.shtml (2009)
Cut-The-Knot: Pythagorean theorem. http://www.cut-the-knot.org/pythagoras/index.shtml (2009)
Gao, X.S., Zhang, J.Z., Chou, S.C.: Geometry Expert. Nine Chapters Pub. (in Chinese) (1998)
Chou, S.C., Gao, X.S., Zhang, J.Z.: An introduction to geometry expert. In: McRobbie, M.A., Slaney, J.K. (eds.) Proc. CADE-13, pp. 235–239. Springer, New York (1996)
Gao, X.S.: Building dynamic mathematical models with geometry expert, III. A geometry deductive database. In: Yang, W., Wang, D. (eds.) Proc. ASCM99. ATCM, Chadstone Centre (1999)
Gao, X.S., Lin, Q.: MMP/Geometer—a software package for automated geometry reasoning. In: Winkler, F. (ed.) Automated Deduction in Geometry, pp. 44–66 (2004)
Chou, S.C., Gao, X.S., Ye, Z.: Java geometry expert server: http://woody.cs.wichita.edu (2009) (One can run the most part of JGEX with a browser.)
Les Editions du Kangourou: Le théorème de Tháles. http://www.mathkang.org/swf/thales2.html (2009)
Chou, S.C., Gao, X.S., Zhang, J.Z.: Machine Proofs in Geometry. World Scientific, Singapore (1994)
Sktektee, S., Jackiw, N., Chanan, S.: Geometer’s Sketchpad Verison 4.0, Reference Manual. Key Curriculum, Emeryville (2001)
Laborde, J.M., et al.: Cabri II Plus (2000)
Kortenkamp, U., Richter-Gebert, J.: User Manual for the Interactive Geometry Software Cinderella. Springer, New York (2000)
Chou, S.C.: Mechanical Geometry Theorem Proving. D. Reidel, Dordrecht (1988)
Taylor, K.B.: Three circles with collinear centres, solution of advanced problem 3887. Am. Math. Mon. 90, 486–487 (1983)
Penrose, R.: Shadows of the Mind. Oxford University Press, Oxford (1994)
Chou, S.C., Gao, X.S., Ye, Z.: Java geometry expert. In: Proceedings of 10th Asian Technology Conference in Mathematics, pp. 78–84 (2005)
Chou, S.C., Gao, X.S.: A Class of Geometry Statments of Constructive Type and Geometry Theorem Proving, TR-89-37. Department of Computer Sciences, University of Texas at Austin, 32 p., November (1989)
Adler, C.F.: Modern Geometry. McGraw-Hill Book, Sydney (1958)
Coxeter, H.S.M., Greitzer, S.L.: Geometry Revisited, (New Mathematical Library). The Mathematical Association of America, New York (1975)
Hadamard, J.: Lecons de Geometrie Elementaire, I. Paris (1931)
Wu, W.T.: On the decision problem and the mechanization of theorem in elementary geometry. Scientia Sinica 21, 159–172 (1978); Automated theorem proving: after 25 years. A.M.S., Contemp. Math. 29, 213–234 (1984)
Wu, W.T.: Basic Principles of Mechanical Theorem Proving in Geometries. Press, Beijing (in Chinese) (1984). (English Version, Springer-Verlag (1993))
Kapur, D.: Geometry theorem proving using Hilbert’s Nullstellensatz. In: Proc. of SYMSAC’86, pp. 202–208. Waterloo (1986)
Chou, S.C., Schelter, W.F.: Proving geometry theorem with rewrite rules. J. Autom. Reason. 4, 253–273 (1986)
Kutzler, B., Stifter, S.: Automated geometry theorem proving using Buchberger’s algorithm. In: Proc. of SYMSAC’86, pp. 209–214. Waterloo (1986)
Gelernter, H., Hanson, J.R., Loveland, D.W.: Empirical explorations of the geometry-theorem proving machine. In: Proc. West. Joint Computer Conf., pp. 143–147 (1960)
Gelernter, H.: Realization of a geometry-theorem proving machine, In: Feigenbaum, E.A., Feldman, J. (eds.) Computers and Thought, pp. 134–152. Mcgraw Hill, London (1963)
Nevins, A.J.: Plane geometry theorem proving using forward chaining. Artif. Intell. 6(1), 1–23 (1975)
Coelho, H., Pereira, L.M.: Automated reasoning in geometry theorem proving with prolog. J. Autom. Reason. 2(4), 329–390 (1986)
Hilbert, D.: Foundations of Geometry, the first edition (in Germany) was published in 1899. Open Court, La Salla (1971)
Chou, S.C., Gao, X.S., Zhang, J.Z.: Automated generation of of readable proofs with geometric invariants, II. Proving theorems with full-angles. J. Autom. Reason. 17, 349–370 (1996)
Chou, S.C., Gao, X.S., Zhang, J.Z.: A collection of 110 geometry theorems and their machine produced proofs using full-angles. WSU technical report (1995)
Chou, S.C., Gao, X.S., Zhang, J.Z.: A deductive database approach to automated geometry theorem proving and discovering. J. Autom. Reas. 25(3), 219–246 (2000)
Jamnik, M.: Mathematical Reasoning with Diagrams: From Intuition to Automation. CSLI, Stanford University, Stanford (2001)
Guilhot, F.: Premiers pas vers un Cours de Géométrie on Coq pour le Lycée. Research Report N° 4893, 44 pages, INRIA (2003)
Dehlinger, C., Dufourd, J.F., Schreck, P.: Higher-order intuitionistic formalization and proofs in Hilbert’s elementary geometry. In: 3rd Int. Workshop on Automated Deduction in Geometry, Zurich, Switzerland, 2000. LNAI, vol. 2061, pp. 306–323. Springer, New York (2001)
Narboux, J.: Mechanical theorem proving in Tarski’s geometry. In: Botana, F., Recio, T. (eds.) Workshop on Automated Deduction in Geometry, Pontevedra, Spain, 2006. LNAI, vol. 4869, pp. 139–156. Springer, New York (2008)
Narboux, J.: A decision procedure for geometry in Coq. In: TPHOL’04 Proceedings. LNCS, vol. 3223. Springer, New York (2004)
Chou, S.C., Ye, Z., Gao, X.S.: Visually Dynamic Presentation of Proofs in Plane. Part 3. Automated Generation of Visually Dynamic Presentations with the Traditional Method and Automated Addition of Auxiliary Geometry Elements (2009, in preparation)
Dolzmann, A., Gilch, L.A.: Generic Hermitian quantifier elimination. In: Campbell, J.A., Buchberger, B. (eds.) Artificial Intelligence and Symbolic Computation: 7th International Conference, AISC 2004, Linz, Austria. Lecture Notes in Computer Science, vol. 3249, pp. 80–93. Springer, Berlin (2004)
Jamnik, M., Bundy, A., Green, I.: On automating diagrammatic proofs of arithmetic arguments, In: Proceedings of International Joint Conference on AI (1997)
Weispfenning, V.: A new approach to quantifier elimination for real algebra. In: Caviness, B.F., Johnson, J.R. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation, pp. 376–392. Springer, Wien (1998)
Winterstein, D., Bundy, A., Gurr,C.: Dr. Doodle: a diagrammatic theorem prover In: Proceedings of 2nd International Joint Conference, Cork, Ireland, pp. 331–335, July 4–8, (2004)
Winterstein, D.: Using diagrammatic reasoning for theorem proving in a continuous domain. Ph.D. thesis, 263 p., The University of Edinburgh (2004)
Ye, Z., Chou, S.C., Gao, X.S.: A Collection of Examples Created with JGEX. In: http://woody.cs.wichita.edu/collection/index.html (2009)
Ye, Z., Chou, S.C., Gao, X.S.: Visually dynamic presentation of proofs in plane, part 2. Automated generation of visually dynamic presentations with the full-angle method and the deductive database method. J. Autom. Reason. (2009, in press)
Author information
Authors and Affiliations
Corresponding author
Additional information
The work reported here was supported by NSF Grant CCR-0201253.
Zheng Ye is on leave from ZJU and working at Wichita State University.
Rights and permissions
About this article
Cite this article
Ye, Z., Chou, SC. & Gao, XS. Visually Dynamic Presentation of Proofs in Plane Geometry. J Autom Reasoning 45, 213–241 (2010). https://doi.org/10.1007/s10817-009-9162-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10817-009-9162-5