1. Hyperbolic Geometry Theorems Of Girolamo Saccheri, SJ Theorems of Girolamo Saccheri, SJ (1667 1733) and his hyperbolic geometry. A Sample of Saccheri s Contribution to the evolution of Non-Euclidean geometry. http://www.faculty.fairfield.edu/jmac/sj/sacflaw/sacther.htm

Theorems of Girolamo Saccheri, S.J. and his hyperbolic geometry A Sample of Saccheri's Contribution to the evolution of Non-Euclidean geometry More can be read concerning Saccheri's contribution to non-Euclidean geometry by viewing Saccheri's Solution to Euclid's BLEMISH The Origins of Non-Euclidean Geometry The two branches of non-Euclidean geometry are associated with Nicolai Lobachevsky and Bernhard Riemann A sample of Saccheri's non-Euclidean geometry By use of similar triangles and congruent parts of similar triangles on the Saccheri quadrilateral, ABDC with AC = BD and A = B = p /2, he establishes his first 32 theorems. Most are too complicated to be treated in a short paper, but here some examples are merely stated, some are illustrated and some are proven. For those proofs which are brief enough to show here, the main steps are indicated and the reader is invited to fill in the missing details of the argument. A century after Saccheri, the geometers, Lobachevsky, Bolyai and Gauss would realize that, by substituting the acute case or the obtuse case for Euclid's postulate Number V, they could create two consistent geometries. In doing so they built on the progress made by Saccheri who had already proven so many of the needed theorems. They were able to create what we recognize today as the "elliptical" and "hyperbolic" non-Euclidean geometries. Most of Saccheri's first 32 theorems can be found in today's non-Euclidean textbooks. Saccheri's theorems are prefaced by " Sac.

2. A Combination Of Nonstandard Analysis And Geometry Theorem Proving, With Applica The theorem prover Isabelle is used to formalise and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia s reasoning is resolutely geometric in nature but being mechanised using only the existing geometry theorem proving (GTP) techniques http://citeseer.nj.nec.com/fleuriot98combination.html

A Combination of Nonstandard Analysis and Geometry Theorem Proving, with Application to Newton's Principia (1998) (Make Corrections) (5 citations) Jacques D. Fleuriot, Lawrence C. Paulson Lecture Notes in Computer Science Home/Search Context Related View or download: dai.ed.ac.uk/daidb/peop princip.ps.gz cl.cam.ac.uk/TechR AMCLTR442.ps.gz Cached: PS.gz PS PDF DjVu ... Help From: cl.cam.ac.uk/users/lcp projects (more) (Enter author homepages) Rate this article: (best) Comment on this article (Enter summary) Abstract: The theorem prover Isabelle is used to formalise and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia's reasoning is resolutely geometric in nature but contains "infinitesimal" elements and the presence of motion that take it beyond the traditional boundaries of Euclidean Geometry. These present difficulties that prevent Newton's proofs from being mechanised using only the existing geometry theorem proving (GTP) techniques. Using concepts from... (Update) Context of citations to this paper: More ...cl.cam.ac.uk Abstract. The approach previously used to mechanise lemmas and Kepler s Law of Equal Areas from Newton s Principia is here used to mechanically reproduce the famous Propositio Kepleriana or Kepler Problem.

3. 2. GEO - A Collection Of Mechanized Geometry Theorem Proofs next up previous Next 5. The Current State Up 4. Two Examples Previous 1. INTPS a 2. GEO - a collection of mechanized geometry theorem proofs. http://www.mathematik.uni-kl.de/~zca/Reports_on_ca/27/paper_html/node11.html

Next: 5. The Current State Up: 4. Two Examples Previous: 1. INTPS - a 2. GEO - a collection of mechanized geometry theorem proofs As a second application of our general framework we collected examples from mechanized geometry theorem proving scattered over several papers mainly of W.-T. Wu, D. Wang, and S.-C. Chou, but also from other sources. The corresponding GEO table contains about 250 records of examples, most of them considered in Chou's elaborated book [ The examples collected so far are related to the coordinate method as driving engine as described in [ ]. The automated proofs may be classified as constructive (yielding rational expressions to be checked for zero equivalence) or equational (yielding a system of polynomials as premise and one or several polynomials as conclusion). To distinguish between the different problem classes we defined a mandatory tag prooftype that must be one of several alternations defined in the Syntax attribute in the corresponding meta sd-file. Extending/modifying this entry modifies the set of valid proof types. Hence the table is open also for new or refined approaches. According to the general theory, see, e.g., [

4. Mechanical Geometry Theorem Proving Feedback Report a problem Satisfaction survey. Mechanical geometry theorem proving. Purchase this Book Purchase this Book. Source, Mathematics http://portal.acm.org/citation.cfm?id=39060&dl=ACM&coll=portal&CFID=11111111&CFT

5. Geometry Theorem List Geometry 1112H Theorem List - Updated 5.25.99. Theorem 1 If two angles are right angles, then they are congruent. Theorem 2 If two angles are straight angles, then they are equal. http://members.aol.com/Joel604/geolist.html

Geometry 11-12H Theorem List - Updated 5.25.99 Theorem 1: If two angles are right angles, then they are congruent. Theorem 2: If two angles are straight angles, then they are equal. Theorem 3: Theorem of Contrapositives: If a conditional statement is true, then the contrapositive is also true. Theorem 4: If angles are supplementary (2 angles whose sum is 180 ) to the same angle, then they are congruent. Theorem 5: If angles are supplementary to congruent angles, then they are congruent. Theorem 6: If angles are complementary (2 angles whose sum is 90 ) to the same angle, then they are congruent. Theorem 7: If angles are complementary to congruent angles, then they are congruent. Theorem 8: If a segment is added to 2 congruent segments, the sums are congruent (Addition Property). Theorem 9: If an angle is added to two congruent angles, the sums are congruent (Addition Property). Theorem 10: If congruent segments are added to congruent segments, the sums are congruent. Theorem 11: If congruent angles are added to congruent segments, the sums are congruent (Addition Property). Theorem 12: If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent (Subtraction Property).

6. Realization Of A Geometry-theorem Proving Machine M. Hadzikadic , F. Lichtenberger , DYY Yun, An application of knowledgebase technology in education a geometry theorem prover, Proceedings of the fifth ACM http://portal.acm.org/citation.cfm?id=216418&dl=ACM&coll=portal&CFID=11111111&CF

7. Geometry Theorem a topic from mathhistory-list geometry theorem. post a message on this topic post a message on a new topic 6 May 1996 geometry theorem http://mathforum.org/epigone/math-history-list/plilgrendski

a topic from math-history-list geometry theorem post a message on this topic post a message on a new topic 6 May 1996 geometry theorem , by 9441749@NESSIE.CC.WWU.EDU 10 May 1996 geometry theorem , by 9441749@NESSIE.CC.WWU.EDU 13 May 1996 Re: geometry theorem , by SIU Man-Keung, HKU The Math Forum

8. Dynamic Geometry Theorem Prover By Jacques Gressier Dynamic geometry theorem prover by Jacques Gressier. reply to this message. post a message on a new topic. Back to geometryannouncements Subject Dynamic geometry theorem prover Author Jacques http://mathforum.com/epigone/geometry-announcements/permpayzhing

Dynamic geometry theorem prover by Jacques Gressier reply to this message post a message on a new topic Back to geometry-announcements Subject: Dynamic geometry theorem prover Author: jacques.gressier@hol.fr Organization: Epigone Date: 21 Jun 1997 11:45:58 -0400 You will find a theorem prover which is the core of a french dynamic geometry software (Windows demo version) at http://wwwperso.hol.fr/~jgressie/index.htm . Much more powerful than SketchPad or Cabri, it can solve any Euclidian geometry problem construction AND proof. ( 75% of the software has been written in Prolog). The teacher can define any new geometry construction exercise. There's also a compiler that can read natural language description of a geometry problem and then generate all possible solutions. The student can try and find the proof. He will be corrected in real time (while defining the figure AND building the proof) and helped to the solution whatever he does. The compiler is not available in the demo version but you can generate construction exercises. We are looking for people who could help us and translate the whole software into other languages. To read articles about experiments in classrooms with this software you can directly go to : http://www.ac-strasbourg.fr/Partenariat/Cari-info/Articles/Anciens/HYPO2COL.htm

9. Geometry Theorem By 9441749 geometry theorem by 9441749. reply to this message post a message on a new topic Back to messages on this topic Back to mathhistory-list next http://mathforum.org/epigone/math-history-list/plilgrendski/Pine.PMDF.3.91.96050

geometry theorem by 9441749 reply to this message post a message on a new topic Back to messages on this topic Back to math-history-list Subject: geometry theorem Author: 9441749@NESSIE.CC.WWU.EDU Date: The Math Forum

10. GEOTHER - Geometry Theorem Prover GEOTHER (geometry theorem provER), a module of Epsilon, is an environment implemented by Dongming Wang in Maple with drawing routines and interface written http://www-calfor.lip6.fr/~wang/GEOTHER/

Geometric Version 1.0 GEOTHER (GEOmetry THeorem provER), a module of Epsilon , is an environment implemented by Dongming Wang in Maple with drawing routines and interface written previously in C and now in Java for manipulating and proving geometric theorems. In GEOTHER a theorem is specified by means of predicates of the form Theorem(H,C,X) asserting that H implies C , where H and C are lists or sets of predicates that correspond to the geometric hypotheses and the conclusion of the theorem, and the optional X is a list of variables used for internal computation. The information contained in the specification may be all that is needed in order to manipulate and prove the theorem. From the specification, GEOTHER can automatically assign coordinates to each point in some optimal manner; translate the predicate representation of the theorem into an English or Chinese statement, into a first-order logical formula, or into algebraic expressions; draw one or several diagrams for the theorem - the drawn diagrams may be animated and modified with a mouse click and dragging, and saved as PostScript files; prove the theorem using any of the five algebraic provers;

11. GEOTHER - Geometry Theorem Prover GEOTHER (geometry theorem provER) is an environment implemented by Dongming Wang in Maple with drawing routines and interface written previously in C and now http://www-calfor.lip6.fr/~wang/epsilon/GEOTHER/

Geometric Version 1.0 GEOTHER (GEOmetry THeorem provER) is an environment implemented by Dongming Wang in Maple with drawing routines and interface written previously in C and now in Java for manipulating and proving geometric theorems. In GEOTHER a theorem is specified by means of predicates of the form Theorem(H,C,X) asserting that H implies C , where H and C are lists or sets of predicates that correspond to the geometric hypotheses and the conclusion of the theorem, and the optional X is a list of variables used for internal computation. The information contained in the specification may be all that is needed in order to manipulate and prove the theorem. From the specification, GEOTHER can automatically assign coordinates to each point in some optimal manner; translate the predicate representation of the theorem into an English or Chinese statement, into a first-order logical formula, or into algebraic expressions; draw one or several diagrams for the theorem - the drawn diagrams may be animated and modified with a mouse click and dragging, and saved as PostScript files; prove the theorem using any of the five algebraic provers;

12. Mechanical Geometry Theorem Proving|KLUWER Academic Publishers Books » Mechanical geometry theorem Proving. Mechanical geometry theorem Proving. Add to cart. by ShangChing Chou Materials Technology http://www.wkap.nl/prod/b/1-4020-0330-7

Title Authors Affiliation ISBN ISSN advanced search search tips Books Mechanical Geometry Theorem Proving Mechanical Geometry Theorem Proving Add to cart by Shang-Ching Chou Materials Technology Laboratory, U.S. Army, Watertown, MA, USA Book Series: MATHEMATICS AND ITS APPLICATIONS Volume 41 Review(s) This work is, in my opinion, completely revolutionary. I believe that, by itself, the book will convince any mathematician in the world that the automation of mathematical reasoning is a profound and rewarding enterprise of extraordinary potential. Robert S. Boyer D. Reidel Publishing Company Hardbound, ISBN 90-277-2650-7 December 1987, 372 pp. EUR 214.00 / USD 235.00 / GBP 148.00 Paperback, ISBN 1-4020-0330-7 November 2001, 372 pp. EUR 75.00 / USD 83.00 / GBP 52.00 Home Help section About Us Contact Us ... Search

13. New(?) Geometry Theorem new(?) geometry theorem. post a message on this topic. post a message on a new topic. 3 Sep 1999 new(?) geometry theorem, by F. Alexander Norman. 3 Sep 1999. Re new(?) geometry theorem, by Antreas http://mathforum.com/epigone/geometry-college/bendskangdwex

a topic from geometry-college new(?) geometry theorem post a message on this topic post a message on a new topic 3 Sep 1999 new(?) geometry theorem , by F. Alexander Norman 3 Sep 1999 Re: new(?) geometry theorem , by Antreas P. Hatzipolakis 5 Sep 1999 Re: new(?) geometry theorem , by Antreas P. Hatzipolakis The Math Forum

14. Mathematics Mechanization|KLUWER Academic Publishers Mathematics Mechanization Mechanical geometry theoremProving, Mechanical Geometry Problem-Solving and Polynomial Equations-Solving. Add to cart. http://www.wkap.nl/prod/b/0-7923-5835-X

Title Authors Affiliation ISBN ISSN advanced search search tips Books Mathematics Mechanization Mathematics Mechanization Mechanical Geometry Theorem-Proving, Mechanical Geometry Problem-Solving and Polynomial Equations-Solving Add to cart by Wu Wen-tsun Mathematics Mechanization Research Center, Institute of Systems Science, Chinese Academy of Sciences, Beijing, PRC Book Series: MATHEMATICS AND ITS APPLICATIONS Volume 489 This book is a collection of essays centred around the subject of mathematical mechanization. It tries to deal with mathematics in a constructive and algorithmic manner so that reasoning becomes mechanical, automated and less laborious. The book is divided into three parts. Part I concerns historical developments of mathematics mechanization, especially in ancient China. Part II describes the underlying principles of polynomial equation-solving, with polynomial coefficients in fields restricted to the case of characteristic 0. Based on the general principle, some methods of solving such arbitrary polynomial systems may be found. This part also goes back to classical Chinese mathematics as well as treating modern works in this field. Finally, Part III contains applications and examples. Audience: This volume will be of interest to research and applied mathematicians, computer scientists and historians in mathematics.

15. GRAMY GRAMY A geometry theorem Prover Capable of Construction. Noboru Matsuda and Kurt VanLehn. Some geometry theorems require construction as a part of the proof. http://www.pitt.edu/~mazda/Doc/JAR04/

GRAMY: A Geometry Theorem Prover Capable of Construction Noboru Matsuda and Kurt VanLehn Intelligent Systems Program, Learning Research and Development Center University of Pittsburgh Journal of Automated Reasoning, 32(1), 3-33 (2004) Abstract Keywords : Automated geometry theorem proving, construction, search control, constraint satisfaction problem, intelligent tutoring system (PDF, 281KB) GRAMY project web cite: Advanced Geometry Intelligent Tutoring System affiliated with CIRCLE

16. Enumerative Real Algebraic Geometry: Theorem 4.4 4.ii.c. Proof of Theorem 4.4. Theorem 4.6 (So9, Theorem 4.2) Let L be a real real (nk)-plane, none of whose Plücker coordinates vanishes. http://www.math.umass.edu/~sottile/pages/ERAG/S4/2.3.html

Next: 4.iii Further Extensions of the Schubert Calculus Up: 4.ii The Special Schubert Calculus Previous: 4.ii.b. The Degrees of Grassmann Varieties 4.ii.c. Proof of Theorem Consider the action of the non-zero real numbers R x on R n t e j t j e j where t is in R x (a non-zero real number) and e e e n is a basis for R n (corresponding to the rows of the n by n identity matrix). Let L be a ( n k )-plane. By ( ), the equation for a k -plane K to meet t L non-trivially is t n n b L b p b K the sum over all b in C n k . For K in X a the sum is over those b below a (including a ), by ( ). Removing the common factor t n n a gives t a b L b p b K The case l i equal to 1 of Theorem is implied by the case a n k n n of the following theorem, as X = Gr( k n Theorem 4.6 , Theorem 4.2]) Let L be a real real ( n k t t t k n k in R x such that for every a in C n k the intersection of the Schubert varieties X a X t L X t L X t a L is transverse (so it contains d a ) points) with all points real. Proof. We induct on m to construct numbers t t t k n k in R x having the property that, for all a in C n k a m , the intersection of the Schubert varieties X a X t L X t L X t m L is transverse (over C ) and each of its d a ) points are real.

17. Abstract: Implementation Of A Geometry Theorem Proving Package In SCRATCHPAD II http://webbler.fhs-hagenberg.ac.at/webbler.exe?database=standort.mdb&getpagename

18. Abstract: A Geometry Theorem Proving Package In SCRATCHPAD II Translate this page A geometry theorem Proving Package in SCRATCHPAD II. Johann Heinzelreiter, Herwig Mayr, K. Kusche, B. Kutzler Proc. Berichte aus http://webbler.fhs-hagenberg.ac.at/webbler.exe?database=standort.mdb&getpagename

19. Foundations Of Geometry: Theorem 2 Theorem 2. If BX. In other words, we re going to hypothesize that Theorem 2 is false and show that leads to a contradiction. II. http://www.doublebit.com/archives/math/sstp1979/foundations/theorem2.htm

Theorem 2 If A, B, and C are three non-collinear points and X is a point of AC, then X is the only point on both AC and BX. Jeff's 1979 Notes I. Given non-collinear points A, B, and C, with a point, X, on AC, assume that there is also a point, Y, on both AC and BX. [In other words, we're going to hypothesize that Theorem 2 is false and show that leads to a contradiction]. II. By Axiom III , points X and Y are contained in only one line, contradicting the hypothesis that the points X and Y are contained in both AC and BX. III. Since the hypothesis is false, Theorem 2 must be true.

20. Foundations Of Geometry: Theorem 1 Theorem 1. If A and C are two points, there is a point P such that A, C, and P are noncollinear. Jeff s 1979 Notes. I. Given points http://www.doublebit.com/archives/math/sstp1979/foundations/theorem1.htm

Theorem 1 If A and C are two points, there is a point P such that A, C, and P are non-collinear. Jeff's 1979 Notes I. Given points A and C, there exists a third point, P [ Axiom II II. Since there is only one line which contains both A and C [ Axiom III ], and since the third point, P, cannot also be contained in this line (otherwise contradicting Axiom II), no line contains A, C, and P. III. By Definition 3 , points A, C, and P are non-collinear. Jeff's 2000 Notes Apparently, Axiom II can be broadly interpreted to "manufacture" a non-collinear point with respect to any given line, which would seem to make this a pretty trivial theorem.

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