## Machine Proofs in Geometry: Automated Production of Readable Proofs for Geometry TheoremsThis book reports recent major advances in automated reasoning in geometry. The authors have developed a method and implemented a computer program which, for the first time, produces short and readable proofs for hundreds of geometry theorems.The book begins with chapters introducing the method at an elementary level, which are accessible to high school students; latter chapters concentrate on the main theme: the algorithms and computer implementation of the method.This book brings researchers in artificial intelligence, computer science and mathematics to a new research frontier of automated geometry reasoning. In addition, it can be used as a supplementary geometry textbook for students, teachers and geometers. By presenting a systematic way of proving geometry theorems, it makes the learning and teaching of geometry easier and may change the way of geometry education. |

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### Contents

The Theory of Machine Proof | 1 |

The Area Method | 51 |

Summary of Chapter 2 | 99 |

4 | 119 |

5 | 130 |

Machine Proof in Solid Geometry | 167 |

Vectors and Machine Proofs | 217 |

Constructive Geometry Statements | 230 |

Summary of Chapter 5 | 253 |

Bibliography | 445 |

List of Symbols | 455 |

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### Common terms and phrases

A B C D A₁ ABCD algebraic Algorithm altitude angle Axiom B₁ basic C₁ called centroid circle circumcenter O A B C collinear configuration Constructive description points coordinates define Definition description points A B C eliminants eliminate points equal equation Example expression Figure foot formula four geometry quantities geometry statement geometry theorems given inter F intersection introduced Lemma line passing LINE U V lratio machine proof Mechanical meet midpoint ndg condition obtain opposite oriented orthocenter PABA PABC PACA PACB parallel parallelogram PBAC PBCB perpendicular plane Proposition prove Pythagoras difference quadrilateral r₁ respectively result SABC SABCD SABD SABP SACD SBCD segments Show sides signed simplify space squares Take a point triangle ABC true VABCD vector vertices volume