## 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. |

### Contents

Foreword | 1 |

Summary of Chapter 2 | 99 |

4 | 119 |

5 | 130 |

Eliminating Points from Geometry Quantities | 145 |

Machine Proof in Solid Geometry | 167 |

Summary of Chapter 4 | 214 |

Constructive Geometry Statements | 230 |

Summary of Chapter 5 | 253 |

214 | 356 |

Bibliography | 445 |

List of Symbols | 455 |

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### Common terms and phrases

ABCD affine geometry altitude Axiom B₁ B₂ bisector C₁ centroid Ceva's theorem circumcenter O A B C circumcircle co-side theorem Constructive description circle Constructive description points D₁ description points A B C description The machine E₁ eliminate points exterior products Figure foot D A B C foot E B A C full-angles geometry quantities geometry theorems given triangle incenter A I B C inter A₁ inter F intersection introduced Lemma line passing line PQ LINE U V lratio M₁ M₂ machine proof midpoint ndg condition O₁ orthic triangle orthocenter P₁ PABA PABC PACA PACB parallel parallelogram PBAC pedal triangle perpendicular plane Proposition Pythagoras difference quadrilateral r₁ S. C. Chou SABA SABB SABC SABCD SABD SABO SABP SACA SACD SACO SACP SBCD SBCE SBCO Show sides BC simplify 1 Example SPQR Take a point three points triangle ABC VABCD vector vertex vertices