Abstract
This paper presents a method of producing readable proofs for theorems in solid geometry. The method is for a class of constructive geometry statements about straight lines, planes, circles, and spheres. The key idea of the method is to eliminate points from the conclusion of a geometric statement using several (fixed) high-level basic propositions about the signed volumes of tetrahedrons and Pythagorean differences of triangles. We have implemented the algorithm, and more than 80 examples from solid geometry have been used to test the program. Our program is efficient and the proofs produced by it are generally short and readable.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Chou, S. C.: Proving elementary geometry theorems using Wu's algorithm,Automated Theorem Proving: After 25 years, Amer. Math. Soc.,Contemporary Mathematics 29 (1984), 243–286.
Chou, S. C.:Mechanical Geometry Theorem Proving, D. Reidel, Dordrecht, 1988.
Chou, S. C., Gao, X. S., and Zhang, J. Z.: Automated production of traditional proofs for constructive geometry theorems, inProc. 8th IEEE Symp. Logic in Computer Science, 1993, pp. 48–56.
Chou, S. C., Gao, X. S., and Zhang, J. Z.: Automated geometry theorem proving using vector calculation, inProc. of ISSAC'93, Kiev, 1993, pp. 284–291.
Chou, S. C., Gao, X. S., and Zhang, J. Z.: Automated production of traditional proofs for theorems in Euclidean geometry, IV. A Collection of 400 Geometry Theorems, TR-92-7, CS Dept., WSU, 1992.
Gelerntner, H., Hanson, J. R. and Loveland, D. W.: Empirical explorations of the geometry-theorem proving machine, inProc. West. Joint Computer Conf., 1960, pp. 143–147.
Hilbert, D.:The Foundations of Geometry, Open Court Pub. Comp., Lasalla, Illinois, 1971.
Kapur, D.: Geometry theorem proving using Hilbert's nullstellensatz, inProc. SYM-SAC'86, Waterloo, 1986, pp. 202–208.
Koedinger, K. R. and Anderson, J. R.: Abstract planning and perceptual chunks: Elements of expertise in geometry,Cognitive Science 14 (1990), 511–550.
Stifter, S.: Geometry theorem proving in vector spaces by means of Gröbner bases, inProc. of ISSAC-93, Kiev, ACM Press, 1993, pp. 301–310.
Mccharen, J. D., Overbeek, R. A., and Wos, L. A.: Problems and experiments for and with Automated Theorem-proving programs,IEEE Trans. on Computers,C-25 (1976), 773–782.
Wu Wen-tsün.: On the Decision Problem and the Mechanization of Theorem in Elementary Geometry,Scientia Sinica 21 (1978), 159–172; also inAutomated Theorem Proving: After 25 years, Contemp. Math.29 A.M.S. 1984, 213–234.
Wu Wen-tsün.: Toward mecahnization of geometry — some comments on Hilbert's “Grundlagen der Geometrie”,Acta Math. Scientia 2 (1982), 125–138.
Zhang, J. Z. and Cao, P. S.:From Education of Mathematics to Mathematics for Education, Sichuan Educational Publishing Inc., 1988 (in Chinese).
Zhang, J. Z., Chou, S. C., and Gao, X. S.: Automated production of traditional proofs for theorems in Euclidean geometry, I. The Hilbert intersection point theorems, TR-92-3, Department of Computer Science, WSU, 1992.Ann. A.I. Math. (to appear).
Author information
Authors and Affiliations
Additional information
The work reported here was supported in part by the NSF Grant CCR-9117870 and Chinese National Science Foundation.
On leave from Institute of Systems Sciences, Academia Sinica, Beijing 100080, P.R. China.
Rights and permissions
About this article
Cite this article
Chou, SC., Gao, XS. & Zhang, JZ. Automated production of traditional proofs in solid geometry. J Autom Reasoning 14, 257–291 (1995). https://doi.org/10.1007/BF00881858
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00881858