Machine proofs in geometry: automated production of readable proofs for geometry theorems
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore u.a.
World Scientific
1994
|
| Series: | Series on applied mathematics
6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 461 S. graph. Darst. |
| ISBN: | 9810215843 |
Staff View
MARC
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| 245 | 1 | 0 | |a Machine proofs in geometry |b automated production of readable proofs for geometry theorems |c Shang-Ching Chou ; Xiao-Shan Gao ; Jing-Zhong Zhang |
| 264 | 1 | |a Singapore u.a. |b World Scientific |c 1994 | |
| 300 | |a XVII, 461 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Series on applied mathematics |v 6 | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Automatic theorem proving | |
| 650 | 4 | |a Axioms |x Data processing | |
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| 700 | 1 | |a Zhang, Jingzhong |e Verfasser |4 aut | |
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Record in the Search Index
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|---|---|
| adam_text | Table of Contents
Foreword vii
Preface ix
Part I. The Theory of Machine Proof
1 Geometry Preliminaries l
1.1 Introduction 1
1.2 Directed Line Segments 3
1.3 Areas and Signed Areas 5
1.4 The Co side Theorem 8
1.5 Parallels 16
1.6 The Co angle Theorem 20
1.7 Pythagoras Differences 26
1.8 Trigonometric Functions 33
1.9 Circles 38
1.10 Full Angles 44
Summary of Chapter 1 49
2 The Area Method 51
2.1 Traditional Proofs Versus Machine Proofs 51
2.2 Signed Areas of Oriented Triangles 54
2.2.1 The Axioms 54
2.2.2 Basic Propositions 57
2.3 The Hilbert Intersection Point Statements 59
2.3.1 Description of the Statements 60
2.3.2 The Predicate Form 63
2.4 The Area Method 64
2.4.1 Eliminating Points from Areas 65
2.4.2 Eliminating Points from Length Ratios 66
xiii
xiv Contrail
2.4.3 Free Points and Area Coordinates 68
2.4.4 Working Examples 71
2.5 More Elimination Techniques 76
2.5.1 Refined Elimination Techniques 76
2.5.2 The Two line Configuration 79
2.6 Area Method and Affine Geometry 81
2.6.1 Affine Plane Geometry 82
2.6.2 Area Method and Affine Geometry 83
2.7 Applications 86
2.7.1 Formula Derivation 86
2.7.2 Existence of n3 Configurations 90
2.7.3 Transversals for Polygons 93
Summary of Chapter 2 99
3 Machine Proof in Plane Geometry 101
3.1 The Pythagoras Difference 101
3.1.1 Pythagoras Difference and Perpendicular 101
3.1.2 Pythagoras Difference and Parallel 103
3.1.3 Pythagoras Difference and Area 105
3.2 Constructive Geometry Statements 107
3.2.1 Linear Constructive Geometry Statements 107
3.2.2 A Minimal Set of Constructions 109
3.2.3 The Predicate Form Ill
3.3 Machine Proof for Class CL 112
3.3.1 The Algorithm 112
3.3.2 Refined Elimination Techniques 117
3.4 The Ratio Constructions 119
3.4.1 More Ratio Constructions 120
3.4.2 Mechanization of Full Angles 125
3.5 Area Coordinates 130
3.5.1 Area Coordinate Systems 130
3.5.2 Area Coordinates and Special Points of Triangles 132
3.6 Trigonometric Functions and Co Circle Points 138
3.6.1 The Co circle Theorems 138
3.6.2 Eliminating Co Circle Points 140
Contents XV
3.7 Machine Proof for Class C 144
3.7.1 Eliminating Points from Geometry Quantities 145
3.7.2 Pseudo Divisions and Triangular Forms 147
3.7.3 Machine Proof for Class C 150
3.8 Geometry Information Bases and Machine Proofs Based on Full Angles . 154
3.8.1 Building the Geometry Information Base 154
3.8.2 Machine Proof Based on the Geometry Information Base 159
Summary of Chapter 3 165
4 Machine Proof in Solid Geometry 167
4.1 The Signed Volume 167
4.1.1 Co face Theorem 169
4.1.2 Volumes and Parallels 171
4.1.3 Volumes and Affine Geometry of Dimension Three 173
4.2 Constructive Geometry Statements 176
4.2.1 Constructive Geometry Statements 177
4.2.2 Constructive Configurations 180
4.3 Machine Proof for Class SH 183
4.3.1 Eliminating Points from Volumes 183
4.3.2 Eliminating Points from Area Ratios 186
4.3.3 Eliminating Points from Length Ratios 188
4.3.4 Free Points and Volume Coordinates 192
4.3.5 Working Examples 194
4.4 Pythagoras Differences in Space 197
4.4.1 Pythagoras Difference and Perpendicularity 197
4.4.2 Pythagoras Difference and Volume 200
4.5 The Volume Method 203
4.5.1 The Algorithm 203
4.5.2 Working Examples 206
4.6 Volume Coordinate System 211
Summary of Chapter 4 214
5 Vectors and Machine Proofs 217
5.1 Metric Vector Spaces of Dimension Three 217
5.1.1 Inner Products and Metric Vector Space 219
5.1.2 Exterior Products in Metric Vector Space 223
xvi Coataab
5.2 The Solid Metric Geometry 225
5.2.1 Inner Products and Cross Products 227
5.2.2 Constructive Geometry Statements 230
5.3 Machine Proof by Vector Calculation 231
5.3.1 Eliminating Points From Vectors 231
5.3.2 Eliminating Points from Inner and Exterior Products 234
5.3.3 TheAlgorithm 236
5.4 Machine Proof in Metric Plane Geometries 239
5.4.1 Vector Approach for Euclidean Plane Geometry 240
5.4.2 Machine Proof in Minkowskian Plane Geometry 243
5.5 Machine Proof Using Complex Numbers 246
Summary of Chapter 5 253
Part II. Topics From Geometry
A Collection of 400 Mechanically Proved Theorems
6 Topics From Geometry 255
6.1 Notation Convention 255
6.2 Geometry of Incidence 259
6.2.1 Menelaus Theorem 259
6.2.2 Ceva s Theorem 262
6.2.3 The Cross ratio and Harmonic Points 266
6.2.4 Pappus Theorem and Desargues Theorem 270
6.2.5 Miscellaneous 274
6.3 Triangles 281
6.3.1 Medians and Centroids 281
6.3.2 Altitudes and Orthocenters 293
6.3.3 The Circumcircle 303
6.3.4 TheEulerLine 321
6.3.5 The Nine Point Circle 323
6.3.6 Incircles and the Excircles 327
6.3.7 Intercept Triangles 341
6.3.8 Equilateral Triangles 346
6.3.9 Pedal Triangles 348
6.3.10 Miscellaneous 350
Contents xvii
6.4 Quadrilaterals 357
6.4.1 General Quadrilaterals 357
6.4.2 Complete Quadrilaterals 365
6.4.3 Parallelograms 369
6.4.4 Squares 380
6.4.5 Cyclic Quadrilaterals 387
6.4.6 Orthodiagonal Quadrilaterals 393
6.4.7 The Butterfly Theorems 396
6.5 Circles 400
6.5.1 Chords, Secants, and Tangents 400
6.5.2 Intersectional Circles 415
6.5.3 The Inversion 421
6.5.4 Orthogonal Circles 426
6.5.5 The Simson Line 429
6.5.6 The Pascal Configuration 435
6.5.7 Cantor s Theorems 439
6.6 A Summary 440
Bibliography 445
List of Symbols 455
Index 457
|
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| author | Chou, Shang-Ching Gao, Xiao-Shan 1963- Zhang, Jingzhong |
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| dewey-ones | 516 - Geometry |
| dewey-raw | 516/.00285/51 |
| dewey-search | 516/.00285/51 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T17:46:08Z |
| institution | BVB |
| isbn | 9810215843 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006689996 |
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| physical | XVII, 461 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | World Scientific |
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| series | Series on applied mathematics |
| series2 | Series on applied mathematics |
| spelling | Chou, Shang-Ching Verfasser aut Machine proofs in geometry automated production of readable proofs for geometry theorems Shang-Ching Chou ; Xiao-Shan Gao ; Jing-Zhong Zhang Singapore u.a. World Scientific 1994 XVII, 461 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Series on applied mathematics 6 Datenverarbeitung Automatic theorem proving Axioms Data processing Geometrie (DE-588)4020236-7 gnd rswk-swf Automatisches Beweisverfahren (DE-588)4069034-9 gnd rswk-swf Automatisches Beweisverfahren (DE-588)4069034-9 s Geometrie (DE-588)4020236-7 s DE-604 Gao, Xiao-Shan 1963- Verfasser (DE-588)121426955 aut Zhang, Jingzhong Verfasser aut Series on applied mathematics 6 (DE-604)BV007228284 6 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006689996&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chou, Shang-Ching Gao, Xiao-Shan 1963- Zhang, Jingzhong Machine proofs in geometry automated production of readable proofs for geometry theorems Series on applied mathematics Datenverarbeitung Automatic theorem proving Axioms Data processing Geometrie (DE-588)4020236-7 gnd Automatisches Beweisverfahren (DE-588)4069034-9 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4069034-9 |
| title | Machine proofs in geometry automated production of readable proofs for geometry theorems |
| title_auth | Machine proofs in geometry automated production of readable proofs for geometry theorems |
| title_exact_search | Machine proofs in geometry automated production of readable proofs for geometry theorems |
| title_full | Machine proofs in geometry automated production of readable proofs for geometry theorems Shang-Ching Chou ; Xiao-Shan Gao ; Jing-Zhong Zhang |
| title_fullStr | Machine proofs in geometry automated production of readable proofs for geometry theorems Shang-Ching Chou ; Xiao-Shan Gao ; Jing-Zhong Zhang |
| title_full_unstemmed | Machine proofs in geometry automated production of readable proofs for geometry theorems Shang-Ching Chou ; Xiao-Shan Gao ; Jing-Zhong Zhang |
| title_short | Machine proofs in geometry |
| title_sort | machine proofs in geometry automated production of readable proofs for geometry theorems |
| title_sub | automated production of readable proofs for geometry theorems |
| topic | Datenverarbeitung Automatic theorem proving Axioms Data processing Geometrie (DE-588)4020236-7 gnd Automatisches Beweisverfahren (DE-588)4069034-9 gnd |
| topic_facet | Datenverarbeitung Automatic theorem proving Axioms Data processing Geometrie Automatisches Beweisverfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006689996&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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