Episodes in Nineteenth and Twentieth Century Euclidean Geometry (Anneli Lax New Mathematical Library, Series Number 37)

Professor Honsberger has succeeded in 'finding' and 'extricating' unexpected and little known properties of such fundamental figures as triangles, results that deserve to be better known. He has laid the foundations for his proofs with almost entirely synthetic methods easily accessible to students of Euclidean geometry early on. While in most of his other books Honsberger presents each of his gems, morsels, and plums, as self contained tidbits, in this volume he connects chapters with some deductive treads. He includes exercises and gives their solutions at the end of the book. In addition to appealing to lovers of synthetic geometry, this book will stimulate also those who, in this era of revitalizing geometry, will want to try their hands at deriving the results by analytic methods. Many of the incidence properties call to mind the duality principle; other results tempt the reader to prove them by vector methods, or by projective transformations, or complex numbers. Cover......Page 1 Title page......Page 4 copyright page......Page 5 1. Cleavers and Splitters......Page 16 2. The Orthocenter......Page 32 3. On Triangles......Page 42 4. On Quadrilaterals......Page 50 2. Cyclic Quadrilaterals......Page 0 Exercise Set 4......Page 56 2. The Simson Line......Page 58 3. The Proof of the Property (John Rigby)......Page 59 4. A Corollary......Page 61 5. A Property of Parabolas......Page 62 6. The Fuhrmann Circle......Page 64 2. Isogonal Lines and Points......Page 68 Exercise......Page 71 3. The Symmedians and the Symmedian Point K......Page 72 4. Applications and Further Developments......Page 74 Exercise Set 7......Page 89 2. The Theorem of Miquel......Page 94 3. The Case of P_1, P_2, P_3 Collinear......Page 96 4. Simson Lines......Page 97 5. A Curious Angle Property......Page 98 1. Parallels and antiparallels......Page 102 2. The Lemoine circles......Page 103 3. The Tucker circles......Page 104 4. The center of a Tucker circle lies on the line KO......Page 107 5. The first Lemoine circle......Page 109 6. The Taylor Circle......Page 110 Exercise Set 9......Page 113 1. The Brocard Points......Page 114 2. The Brocard Angle......Page 116 Exercise......Page 119 3. The Brocard Circle......Page 121 4. The Brocard triangles......Page 125 5. The Steiner point and the Tarry point......Page 134 6. A property relating K, G, Omega, Omega'......Page 136 Section 1......Page 140 Section 2......Page 142 3. The Rigby Point......Page 147 Exercise......Page 151 1. Ceva’s Theorem......Page 152 Section 2......Page 153 Section 3......Page 156 4. Haruki’s Cevian theorem for circles......Page 159 Section 1......Page 162 2. Applications......Page 164 Suggested Reading......Page 170 1. Cleavers and Splitters......Page 172 3. On Triangles......Page 174 4. On Quadrilaterals......Page 175 7. The Symmedian Point......Page 177 9. The Tucker Circles......Page 185 11. The Orthopole......Page 187 Index......Page 188 "Professor Honsberger has succeeded in "finding" and "extricating" unexpected and little known properties of such fundamental figures as triangles, results that deserve to be better known. He has laid the foundations for his proofs with almost entirely synthetic methods easily accessible to students of Euclidean geometry early on. He includes exercises and gives their solutions at the end of the book." "In addition to appealing to lovers of synthetic geometry, this book will stimulate also those who, in this era of revitalizing geometry, will want to try their hands at deriving the results by analytic methods. Many of the incidence properties call to mind the duality principle; other results tempt the reader to prove them by vector methods, or by projective transformations, or complex numbers."--Jacket Today's personal computer gives its owner tremendous power which can be used for experimental investigations and simulations of unprecedented scope, leading to mini-research. This book is a first step into this exciting field. This is a mathematics book, not a programming book, although it explains Pascal to beginners. It is aimed at high school students and undergraduates with a strong interest in mathematics and teachers looking for fresh ideas. It is full of diverse mathematical ideas requiring little background. It includes a large number of challenging problems that illustrate how computing leads to conjectures, many of which can then be proved by mathematical reasoning. - Back cover Euclidean geometry was worked out by Euclid and his predecessors more than 2300 years ago and is studied today mostly as a background to other branches of mathematics. In fact, however, as Professor Honsberger masterfully demonstrates, geometry in the style of Euclid is still alive and well. Mathematicians have again been studying the properties of geometric figures from a synthetic point of view and have discovered many new and unexpected results which Euclid himself never found. And since all of us have studied Euclidean geometry, at least the ancient version, this book is easily accessible. Exercises with their solutions are included in the book Euclidean geometry Euclidean,geometry;,geometry;,Brocard,points;,cevians;,orthocenter;,theorem,of,Menelaus Euclidean geometry,geometry,Brocard points,cevians,orthocenter,theorem of Menelaus Presents topology as a unifying force for larger areas of mathematics through its application in existence theorems.