## Geometry: Our Cultural HeritageThis book is based on lectures on geometry at the University of Bergen, Norway. Over the years these lectures have covered many different aspects and facets ofthis wonderful field.Consequently it has ofcourse never been possible to give a full and final account ofgeometry as such, at an undergraduate level: A carefully considered selection has always been necessary.The present book constitutes the main central themes of these selections. One of the groups I am aiming at, is future teachers of mathematics. All too often the geometry which goes into the syllabus for teacher-students present the material as pedantic and formalistic, suppressing the very pow erful and dynamic character of this old - and yet so young! - field. A field of mathematical insight, research, history and source of artistic inspiration. And not least important, a foundation for our common cultural heritage. Another motivation is to provide an invitation to mathematics in gen eral. It is an unfortunate fact that today, at a time when mathematics and knowledge of mathematics is more important than ever, phrases like math avoidance and math anxiety are very much in the public vocabulary. An im portant task is seriously attempting to heal these ills. Ills perhaps inflicted on students at an early age, through deficient or even harmful teaching prac tices. Thus the book also aims at an informed public, interested in making a new beginning in math. And in doing so, learning more about this part of our cultural heritage. |

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### Contents

II | 3 |

III | 4 |

IV | 7 |

V | 11 |

VI | 12 |

VII | 15 |

VIII | 27 |

IX | 29 |

LVI | 201 |

LVII | 207 |

LVIII | 211 |

LIX | 212 |

LX | 214 |

LXI | 221 |

LXII | 222 |

LXIII | 224 |

X | 38 |

XI | 40 |

XII | 45 |

XIII | 48 |

XIV | 50 |

XV | 51 |

XVI | 54 |

XVII | 56 |

XVIII | 59 |

XIX | 67 |

XX | 69 |

XXI | 76 |

XXII | 78 |

XXIII | 93 |

XXIV | 95 |

XXV | 102 |

XXVI | 106 |

XXVII | 111 |

XXVIII | 113 |

XXIX | 115 |

XXX | 119 |

XXXI | 120 |

XXXII | 126 |

XXXIII | 127 |

XXXIV | 131 |

XXXV | 134 |

XXXVI | 135 |

XXXVII | 138 |

XXXVIII | 139 |

XXXIX | 140 |

XL | 144 |

XLI | 151 |

XLII | 153 |

XLIII | 156 |

XLIV | 159 |

XLV | 165 |

XLVI | 167 |

XLVII | 168 |

XLVIII | 170 |

XLIX | 171 |

L | 176 |

LI | 177 |

LII | 182 |

LIII | 185 |

LIV | 195 |

LV | 200 |

LXIV | 225 |

LXV | 233 |

LXVII | 235 |

LXVIII | 236 |

LXX | 237 |

LXXI | 245 |

LXXII | 248 |

LXXIII | 254 |

LXXIV | 257 |

LXXV | 261 |

LXXVI | 269 |

LXXVII | 270 |

LXXVIII | 274 |

LXXIX | 277 |

LXXX | 281 |

LXXXI | 282 |

LXXXII | 285 |

LXXXIII | 287 |

LXXXIV | 296 |

LXXXV | 300 |

LXXXVI | 301 |

LXXXVII | 304 |

LXXXVIII | 307 |

LXXXIX | 308 |

XC | 311 |

XCI | 314 |

XCII | 316 |

XCIII | 318 |

XCIV | 322 |

XCV | 329 |

XCVI | 330 |

XCVII | 331 |

XCVIII | 338 |

XCIX | 339 |

C | 340 |

CI | 348 |

CII | 353 |

CIII | 354 |

CIV | 355 |

CV | 356 |

CVI | 357 |

CVII | 359 |

CVIII | 360 |

363 | |

365 | |

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### Common terms and phrases

affine affine transformation Alexandria algebraic angle Archimedes assume axioms base called circle claim common compass complete compute concept cone conic section consider consists construction contained coordinate system corresponding course curve defined definition denote Desargues determined distance draw elements equal equation equivalent Euclid's Euclidian example exist explained expressed fact Figure finally fixed formula four geometry give given Greek hyperbola important infinity integers intersection Italy known later length mathematician mathematics means multiplicity namely natural objects obtained origin passing Persian plane polynomial positive possible Postulate present problem projective proof Proposition prove Pythagoras Pythagoreans referred regular relation result Roman root rule segments shown side simple situation space sphere square statement straight straightedge surface Theorem theory triangle yields zero