## Mechanical Theorem Proving in Geometries: Basic PrinciplesThere seems to be no doubt that geometry originates from such practical activ ities as weather observation and terrain survey. But there are different manners, methods, and ways to raise the various experiences to the level of theory so that they finally constitute a science. F. Engels said, "The objective of mathematics is the study of space forms and quantitative relations of the real world. " Dur ing the time of the ancient Greeks, there were two different methods dealing with geometry: one, represented by the Euclid's "Elements," purely pursued the logical relations among geometric entities, excluding completely the quantita tive relations, as to establish the axiom system of geometry. This method has become a model of deduction methods in mathematics. The other, represented by the relevant work of Archimedes, focused on the study of quantitative re lations of geometric objects as well as their measures such as the ratio of the circumference of a circle to its diameter and the area of a spherical surface and of a parabolic sector. Though these approaches vary in style, have their own features, and reflect different viewpoints in the development of geometry, both have made great contributions to the development of mathematics. The development of geometry in China was all along concerned with quanti tative relations. |

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### Contents

Desarguesian geometry and the Desarguesian number system | 13 |

12 The axiom of infinity and Desargues axioms | 18 |

13 Rational points in a Desarguesian plane | 25 |

14 The Desarguesian number system and rational number subsystem | 30 |

15 The Desarguesian number system on a line | 37 |

16 The Desarguesian number system associated with a Desarguesian plane | 42 |

17 The coordinate system of Desarguesian plane geometry | 55 |

Orthogonal geometry metric geometry and ordinary geometry | 63 |

42 Factorization of polynomials | 152 |

43 Wellordering of polynomial sets | 159 |

44 A constructive theory of algebraic varieties irreducible ascending sets and irreducible algebraic varieties | 169 |

45 A constructive theory of algebraic varieties irreducible decomposition of algebraic varieties | 178 |

46 A constructive theory of algebraic varieties the notion of dimension and the dimension theorem | 183 |

47 Proof of the mechanization theorem of unordered geometry | 187 |

48 Examples for the mechanical method of unordered geometry | 195 |

Mechanization theorems of ordinary ordered geometries | 213 |

22 Orthogonal axioms and unordered orthogonal geometry | 70 |

23 The orthogonal coordinate system of unordered orthogonal geometry | 80 |

24 Unordered metric geometry | 91 |

25 The axioms of order and ordered metric geometry | 102 |

26 Ordinary geometry and its subordinate geometries | 109 |

Mechanization of theorem proving in geometry and Hilberts mechanization theorem | 115 |

32 The standardization of coordinate representation of geometric concepts | 118 |

33 The mechanization of theorem proving and Hilberts mechanization theorem about pure point of intersection theorems in Pascalian geometry | 124 |

34 Examples for Hilberts mechanical method | 128 |

35 Proof of Hilberts mechanization theorem | 139 |

The mechanization theorem of ordinary unordered geometry | 149 |

52 Tarskis theorem and Seidenbergs method | 220 |

53 Examples for the mechanical method of ordered geometries | 228 |

Mechanization theorems of various geometries | 235 |

62 The mechanization of theorem proving in projective geometry | 236 |

63 The mechanization of theorem proving in BolyaiLobachevskys hyperbolic nonEuclidean geometry | 246 |

64 The mechanization of theorem proving in Riemanns elliptic nonEuclidean geometry | 258 |

65 The mechanization of theorem proving in two circle geometries | 264 |

66 The mechanization of formula proving with transcendental functions | 267 |

281 | |

285 | |

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### Common terms and phrases

according to Sect algebraic variety arbitrary point ascending set axiom for intersecting axiom of infinity axiom of parallels axiom system axioms of order basic set called Chap circle coincide collinear concept congruence consider construct decomposition define definition degenerate denoted Desargues Desarguesian number system Desarguesian plane determined equations Euclidean geometry example extended zero extension field finite number fundamental objects fundamental relations geometric theorem Hence Hilbert's axioms hypothesis intersecting lines intersection point introduce irreducible polynomial isomorphism correspondences isotropic line Lemma mechanical method mechanizable meeting metric geometry midpoint non-degeneracy conditions non-Euclidean geometry non-isotropic line number field number of steps ordered geometries ordinary geometry orthocenter orthogonal coordinate orthogonal geometry orthogonal rate parallelogram Pascalian axiom Pascalian geometry points and lines polynomial set problem projective geometry proof proving in geometry real closed field satisfy segment sides Suppose symmetric axes symmetric point triangle unordered