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The Geometry of Homological Triangles

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The Geometry of Homological Triangles FLORENTIN SMARANDACHE ION PĂTRAŞCU THE GEOMETRY OF HOMOLOGICAL TRIANGLES 2012 1 Table of Contents Preface ..………………………………………………………………………………………………………………………………………………….5 Chapter 1 ....................................................................................................................................................... 6 Remarkable pairs of homological triangles .................................................................................................. 6 1.1. Homological triangles’ theorem ......................................................................................................... 6 1.2. Some remarkable homological triangles .......................................................................................... 15 A. The orthic triangle .......................................................................................................................... 15 B. The Cevian triangle ........................................................................................................................ 17 C. The anti-supplemental triangle ....................................................................................................... 18 D. The K-symmedian triangle ............................................................................................................. 19 E. The tangential triangle .................................................................................................................... 23 F. The contact triangle ........................................................................................................................ 24 G. The medial triangle ........................................................................................................................ 27 H. The cotangent triangle .................................................................................................................... 31 I. The ex-tangential triangle ................................................................................................................ 41 J. The circum-pedal triangle (or meta-harmonic) ............................................................................... 43 K. The Coşniţă triangle ....................................................................................................................... 46 Chapter 2 ..................................................................................................................................................... 52 Triplets of homological triangles ................................................................................................................ 52 2.1. Theorems relative to the triplets of homological triangles ............................................................... 52 2.2. A remarkable triplet of homological triangles ................................................................................. 57 2.3. Other theorems on homological triangles ........................................................................................ 72 Chapter 3 ..................................................................................................................................................... 82 Bi-homological and tri-homological triangles ............................................................................................. 82 3.1. The necessary and sufficient condition of homology ...................................................................... 82 3.2. Bi-homological and tri-homological triangles ................................................................................. 83 1.3. Tri-homological equilateral triangles which have the same center ............................................. 91 1.4. The Pappus’ Theorem ................................................................................................................. 93 1.5. The duality principle ................................................................................................................... 94 Chapter 4 ..................................................................................................................................................... 99 Homological triangles inscribed in circle .................................................................................................... 99 4.1. Theorems related to circles. ............................................................................................................. 99 3 4.1. Homological triangles inscribed in a circle. ................................................................................... 106 Chapter 5 ................................................................................................................................................... 116 Proposed problems. Open problems ........................................................................................................ 116 5.1. Proposed problems ......................................................................................................................... 116 5.2. Open problems ............................................................................................................................... 132 Chapter 6 ................................................................................................................................................... 134 Notes ......................................................................................................................................................... 134 6.1. Menelaus’ theorem and Ceva’s theorem ........................................................................................ 134 Annex 1 ..................................................................................................................................................... 139 Important formulae in the triangle geometry ...................................................................................... 139 C. Distances between remarkable points in triangle geometry ........................................................... 140 D. Application ........................................................................................................................................ 142 6.3. The point’s power in rapport to a circle ......................................................................................... 144 6.4. The non-harmonic rapport ............................................................................................................. 150 6.5. Pole and polar in rapport with a circle ........................................................................................... 157 Transformation by duality in rapport with a circle ........................................................................... 159 6.6. Homothety ...................................................................................................................................... 166 Homothety definition ....................................................................................................................... 166 Properties .......................................................................................................................................... 166 Applications ....................................................................................................................................... 170 6.7. Inversion in plane ........................................................................................................................... 175 A. Definition, Notations ................................................................................................................. 175 B. The image of a figure through an inversion .............................................................................. 175 C. The construction with the compass and the ruler of the inverses of a line and of a circle ........ 179 D. Other properties of the inversion .............................................................................................. 183 Applications ....................................................................................................................................... 186 Chapter 7 ................................................................................................................................................... 189 Solutions and indications to the proposed problems ............................................................................... 189 REFERENCES .............................................................................................................................................. 242 4 PREFACE This book is addressed to students, professors and researchers of geometry, who will find herein many interesting and original results. The originality of the book The Geometry of Homological Triangles consists in using the homology of triangles as a “filter” through which remarkable notions and theorems from the geometry of the triangle are unitarily passed. Our research is structured in seven chapters, the first four are dedicated to the homology of the triangles while the last ones to their applications. In the first chapter one proves the theorem of homological triangles (Desargues, 1636), one survey the remarkable pairs of homological triangles, making various connections between their homology centers and axes. Second chapter boards the theorem relative to the triplets of homological triangles. The Veronese Theorem is proved and it is mentioned a remarkable triplet
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