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| Preface | |
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| Straightedge and compass | |
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| Euclid's construction axioms | |
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| Euclid's construction of the equilateral triangle | |
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| Some basic constructions | |
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| Multiplication and division | |
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| Similar triangles | |
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| Discussion | |
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| Euclid's approach to geometry | |
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| The parallel axiom | |
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| Congruence axioms | |
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| Area and equality | |
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| Area of parallelograms and triangles | |
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| The Pythagorean theorem | |
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| Proof of the Thales theorem | |
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| Angles in a circle | |
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| The Pythagorean theorem revisited | |
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| Discussion | |
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| Coordinates | |
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| The number line and the number plane | |
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| Lines and their equations | |
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| Distance | |
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| Intersections of lines and circles | |
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| Angle and slope | |
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| Isometries | |
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| The three reflections theorem | |
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| Discussion | |
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| Vectors and Euclidean spaces | |
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| Vectors | |
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| Direction and linear independence | |
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| Midpoints and centroids | |
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| The inner product | |
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| Inner product and cosine | |
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| The triangle inequality | |
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| Rotations, matrices, and complex numbers | |
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| Discussion | |
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| Perspective | |
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| Perspective drawing | |
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| Drawing with straightedge alone | |
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| Projective plane axioms and their models | |
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| Homogeneous coordinates | |
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| Projection | |
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| Linear fractional functions | |
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| The cross-ratio | |
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| What is special about the cross-ratio? | |
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| Discussion | |
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| Projective planes | |
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| Pappus and Desargues revisited | |
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| Coincidences | |
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| Variations on the Desargues theorem | |
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| Projective arithmetic | |
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| The field axioms | |
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| The associative laws | |
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| The distributive law | |
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| Discussion | |
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| Transformations | |
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| The group of isometries of the plane | |
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| Vector transformations | |
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| Transformations of the projective line | |
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| Spherical geometry | |
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| The rotation group of the sphere | |
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| Representing space rotations by quaternions | |
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| A finite group of space rotations | |
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| The groups S[superscript 3] and RP[superscript 3] | |
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| Discussion | |
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| Non-Euclidean geometry | |
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| Extending the projective line to a plane | |
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| Complex conjugation | |
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| Reflections and Mobius transformations | |
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| Preserving non-Euclidean lines | |
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| Preserving angle | |
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| Non-Euclidean distance | |
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| Non-Euclidean translations and rotations | |
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| Three reflections or two involutions | |
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| Discussion | |
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| References | |
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| Index | |