## Lines and Curves: A Practical Geometry Handbook"Lines and Curves" is a unique adventure in the world of geometry. Originally written in Russian and used in the Gelfand Correspondence School, this work has since become a classic: unlike standard textbooks that use the subject primarily to introduce axiomatic reasoning through formal geometric proofs, "Lines and Curves" maintains mathematical rigor, but also strikes a balance between creative storytelling and surprising examples of geometric properties. This newly revised and expanded edition includes more than 200 theoretical and practical problems in which formal geometry provides simple and elegant insight, and the book points the reader toward important areas of modern mathematics. One of the key strengths of the text is its reinterpretation of geometry in the context of motion, whereby curves are realized as trajectories of moving points instead of as stationary configurations in the plane. This novel approach, rooted in physics and kinematics, yields unusually intuitive and straightforward proofs of many otherwise difficult results. The geometrical properties of paths traced by moving points, the sets of points satisfying given geometric constraints, and questions of maxima and minima are all emphasized; therefore, "Lines and Curves" is well positioned for companion use with software packages like "The Geometer’s SketchpadŽ," and it can serve as a guidebook for engineers. Its deeper, interdisciplinary treatment is ideal for more theoretical readers, and the development from first principles makes the book accessible to undergraduates, advanced high school students, teachers, and puzzle enthusiasts alike. A wide audience will profit from this clear and diverse examination of the subject. |

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### Contents

Sets of Points | 7 |

A family of lines and motion | 11 |

Construction problems | 12 |

Additional problems | 15 |

The Alphabet | 19 |

A circle and a pair of arcs | 21 |

Squares of distances | 23 |

Distances from straight lines | 29 |

Addition of rotations | 91 |

A theorem on two circles | 99 |

Velocities and tangents | 101 |

Parametric equations | 106 |

Conclusion | 109 |

Drawings Animation and the Magic Triangle | 111 |

The envelope of a family of lines | 114 |

The magic triangle | 115 |

The entire alphabet | 33 |

Logical Combinations | 35 |

Intersections and unions | 39 |

The cheese problem | 44 |

Maximum and Minimum | 47 |

Where to put the point? | 49 |

The motorboat problem | 51 |

Level Curves | 55 |

Functions on a plane | 57 |

The graph of a function | 58 |

The map of a function | 62 |

Extrema of functions | 64 |

Quadratic Curves | 67 |

Foci and tangents | 70 |

The focal property of a parabola | 72 |

Curves as the envelopes of straight lines | 75 |

Equations of curves | 78 |

A note about the elimination of radicals | 81 |

The end of our alphabet | 82 |

Algebraic curves | 87 |

Rotations and Trajectories | 89 |

A tiny ring on a circle | 117 |

Two pedestrians on a circle and the Steiner deltoid | 118 |

How three points move around three symmetric circles | 120 |

The rotation of the WallaceSimson line and the Feuerbach circle | 124 |

Steiners triangle and Morleys triangle | 125 |

Answers Hints Solutions | 127 |

Summary of Results from Analytic Geometry | 135 |

Some Facts from School Geometry | 137 |

B2 Distances perpendiculars | 138 |

B3 The circle | 139 |

A Dozen Assignments | 141 |

C2 Transformations and constructions | 142 |

C4 Straight lines and linear relations | 143 |

C6 Partitions | 144 |

C8 Envelopes infinite unions | 145 |

C10 Equations of curves | 146 |

C12 Small investigations | 148 |

149 | |

About Victor Gutenmacher | 153 |

About N B Vasilyev | 155 |

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Lines and Curves: A Practical Geometry Handbook Victor Gutenmacher,N.B. Vasilyev Limited preview - 2013 |