• Medians, centroid, and center of gravity of a system of points
• Altitudes and the orthocenter of a triangle and some of its properties
• The orthic triangle and some of its properties
• The angle bisector of a triangle and its properties
• The area of a quadrilateral
• The theorem of ratios of the areas of similar polygons
• A pivotal approach: applying rotation in problem solving
• Auxiliary elements in problem solving
• Constructions siblings
• Session of one interesting construction problem
• Morley's theorem.

This volume contains a collection of papers based on lectures delivered by distinguished mathematicians at Clay Mathematics Institute events over the past few years. It is intended to be the first in an occasional series of volumes of CMI lectures. Although not explicitly linked, the topics in this inaugural volume have a common flavour and a common appeal to all who are interested in recent developments in geometry. They are intended to be accessible to all who work in this general area, regardless of their own particular research interests.
(source: Nielsen Book Data)9780198784913 20170919

• The Origin of Geometry in Design-- A Constructive Approach to the Pythagorean Theorem-- Lines and Pixels-- Compass and Straightedge Constructions-- Congruent Triangles and Trigonometry-- The Art of Proof-- Parallel Lines and Bracing of Frameworks-- Perpendicular Lines and Vornoi Domains-- Doing Algebra with Geometry-- Areas, Vectors and Geoboards-- From Right Triangles to Logarithmic Spirals-- The Golden and Silver Means-- Transformational Geometry and Isometries-- Kaleidoscope and Frieze Symmetry-- An Introduction to Symmetry Groups-- Fractals, Isometries and Matrices-- Thirteen Fundamental Constructions of Projective Geometry.
• (source: Nielsen Book Data)9789814556705 20160618

This book aims to make the subject of geometry and its applications understandable to first year students majoring in architecture, design, or the liberal arts. It can also be used to teach students at different levels of computational ability, and there is sufficient novel material to interest students of mathematics or engineering keen in improving their visual understanding. While the book rigorously develops the subject of geometry, it also goes deeply into the applications of the subject and contains much introductory material of which the student may not be knowledgeable. The constructive approach, using compass and straightedge, engages students not just on an intellectual level, but also at a tactile level. This may be the only rigorous book on geometry that attempts to reach students outside of the discipline of mathematics.
(source: Nielsen Book Data)9789814556705 20160618

• Preface-- 1. Euclidean geometry-- 2. Axiomatic systems-- 3. Analytic geometry-- 4. Non-Euclidean geometries-- 5. Transformational geometry-- 6. Symmetry-- 7. Projective geometry-- 8. Finite geometries-- 9. Differential geometry-- 10. Discrete geometry-- 11. Epilogue-- Appendix A. Definitions, postulates, common notions, and propositions from Book I of Euclid's Elements-- Appendix B. SMSG axioms for Euclidean geometry-- Appendix C. Hilbert's axioms for Euclidean plane geometry-- Appendix D. Linear algebra summary-- Appendix E. Multivariable calculus summary-- Appendix F. Elements of proofs-- Answers to selected exercises-- Acknowledgements-- Index.
• (source: Nielsen Book Data)9781939512086 20160830

This is a self-contained, comprehensive survey of college geometry that can serve a wide variety of courses for students of both mathematics and mathematics education. The text develops visual insights and geometric intuition while stressing the logical structure, historical development, and deep interconnectedness of the ideas. Chapter topics include Euclidean geometry, axiomatic systems and models, analytic geometry, transformational geometry, symmetry, non-Euclidean geometry, projective geometry, finite geometry, differential geometry, and discrete geometry. The different chapters are as independent as possible, while the text still manages to highlight the many connections between topics. Appendices include material from Euclid's first book, as well as Hilbert's axioms, and provide brief summaries of the parts of linear algebra and multivariable calculus needed for certain chapters.
(source: Nielsen Book Data)9781939512086 20160830

Projective geometry is concerned with the properties of figures that are invariant by projecting and taking sections. It is considered one of the most beautiful parts of geometry and plays a central role because its specializations cover the whole of the affine, Euclidean and non-Euclidean geometries. The natural extension of projective geometry is projective algebraic geometry, a rich and active field of research. Regarding its applications, results and techniques of projective geometry are today intensively used in computer vision. This book contains a comprehensive presentation of projective geometry, over the real and complex number fields, and its applications to affine and Euclidean geometries. It covers central topics such as linear varieties, cross ratio, duality, projective transformations, quadrics and their classifications – projective, affine and metric –, as well as the more advanced and less usual spaces of quadrics, rational normal curves, line complexes and the classifications of collineations, pencils of quadrics and correlations. Two appendices are devoted to the projective foundations of perspective and to the projective models of plane non-Euclidean geometries. The presentation uses modern language, is based on linear algebra and provides complete proofs. Exercises are proposed at the end of each chapter; many of them are beautiful classical results. The material in this book is suitable for courses on projective geometry for undergraduate students, with a working knowledge of a standard first course on linear algebra. The text is a valuable guide to graduate students and researchers working in areas using or related to projective geometry, such as algebraic geometry and computer vision, and to anyone wishing to gain an advanced view on geometry as a whole.

• Preface v PART I EUCLIDEAN GEOMETRY 1 Congruency 3 1.1 Introduction 3 1.2 Congruent Figures 6 1.3 Parallel Lines 12 1.3.1 Angles in a Triangle 13 1.3.2 Thales' Theorem 14 1.3.3 Quadrilaterals 17 1.4 More About Congruency 21 1.5 Perpendiculars and Angle Bisectors 24 1.6 Construction Problems 28 1.6.1 The Method of Loci 31 1.7 Solutions to Selected Exercises 33 1.8 Problems 38 2 Concurrency 41 2.1 Perpendicular Bisectors 41 2.2 Angle Bisectors 43 2.3 Altitudes 46 2.4 Medians 48 2.5 Construction Problems 50 2.6 Solutions to the Exercises 54 2.7 Problems 56 3 Similarity 59 3.1 Similar Triangles 59 3.2 Parallel Lines and Similarity 60 3.3 Other Conditions Implying Similarity 64 3.4 Examples 67 3.5 Construction Problems 75 3.6 The Power of a Point 82 3.7 Solutions to the Exercises 87 3.8 Problems 90 4 Theorems of Ceva and Menelaus 95 4.1 Directed Distances, Directed Ratios 95 4.2 The Theorems 97 4.3 Applications of Ceva's Theorem 99 4.4 Applications of Menelaus' Theorem 103 4.5 Proofs of the Theorems 115 4.6 Extended Versions of the Theorems 125 4.6.1 Ceva's Theorem in the Extended Plane 127 4.6.2 Menelaus' Theorem in the Extended Plane 129 4.7 Problems 131 5 Area 133 5.1 Basic Properties 133 5.1.1 Areas of Polygons 134 5.1.2 Finding the Area of Polygons 138 5.1.3 Areas of Other Shapes 139 5.2 Applications of the Basic Properties 140 5.3 Other Formulae for the Area of a Triangle 147 5.4 Solutions to the Exercises 153 5.5 Problems 153 6 Miscellaneous Topics 159 6.1 The Three Problems of Antiquity 159 6.2 Constructing Segments of Specific Lengths 161 6.3 Construction of Regular Polygons 166 6.3.1 Construction of the Regular Pentagon 168 6.3.2 Construction of Other Regular Polygons 169 6.4 Miquel's Theorem 171 6.5 Morley's Theorem 178 6.6 The Nine-Point Circle 185 6.6.1 Special Cases 188 6.7 The Steiner-Lehmus Theorem 193 6.8 The Circle of Apollonius 197 6.9 Solutions to the Exercises 200 6.10 Problems 201 PART II TRANSFORMATIONAL GEOMETRY 7 The Euclidean Transformations or Isometries 207 7.1 Rotations, Reflections, and Translations 207 7.2 Mappings and Transformations 211 7.2.1 Isometries 213 7.3 Using Rotations, Reflections, and Translations 217 7.4 Problems 227 8 The Algebra of Isometries 231 8.1 Basic Algebraic Properties 231 8.2 Groups of Isometries 236 8.2.1 Direct and Opposite Isometries 237 8.3 The Product of Reflections 241 8.4 Problems 246 9 The Product of Direct Isometries 253 9.1 Angles 253 9.2 Fixed Points 255 9.3 The Product of Two Translations 256 9.4 The Product of a Translation and a Rotation 257 9.5 The Product of Two Rotations 260 9.6 Problems 263 10 Symmetry and Groups 269 10.1 More About Groups 269 10.1.1 Cyclic and Dihedral Groups 273 10.2 Leonardo's Theorem 277 10.3 Problems 281 11 Homotheties 287 11.1 The Pantograph 287 11.2 Some Basic Properties 288 11.2.1 Circles 291 11.3 Construction Problems 293 11.4 Using Homotheties in Proofs 298 11.5 Dilatation 302 11.6 Problems 304 12 Tessellations 311 12.1 Tilings 311 12.2 Monohedral Tilings 312 12.3 Tiling with Regular Polygons 317 12.4 Platonic and Archimedean Tilings 323 12.5 Problems 330 PART III INVERSIVE AND PROJECTIVE GEOMETRIES 13 Introduction to Inversive Geometry 337 13.1 Inversion in the Euclidean Plane 337 13.2 The Effect of Inversion on Euclidean Properties 343 13.3 Orthogonal Circles 351 13.4 Compass-Only Constructions 360 13.5 Problems 369 14 Reciprocation and the Extended Plane 373 14.1 Harmonic Conjugates 373 14.2 The Projective Plane and Reciprocation 383 14.3 Conjugate Points and Lines 393 14.4 Conics 399 14.5 Problems 406 15 Cross Ratios 409 15.1 Cross Ratios 409 15.2 Applications of Cross Ratios 420 15.3 Problems 429 16 Introduction to Projective Geometry 433 16.1 Straightedge Constructions 433 16.2 Perspectivities and Projectivities 443 16.3 Line Perspectivities and Line Projectivities 448 16.4 Projective Geometry and Fixed Points 448 16.5 Projecting a Line to Infinity 451 16.6 The Apollonian Definition of a Conic 455 16.7 Problems 461 Bibliography 464 Index 469.
• (source: Nielsen Book Data)9781118679197 20180530

Features the classical themes of geometry with plentiful applications in mathematics, education, engineering, and science Accessible and reader-friendly, Classical Geometry: Euclidean, Transformational, Inversive, and Projective introduces readers to a valuable discipline that is crucial to understanding bothspatial relationships and logical reasoning. Focusing on the development of geometric intuitionwhile avoiding the axiomatic method, a problem solving approach is encouraged throughout. The book is strategically divided into three sections: Part One focuses on Euclidean geometry, which provides the foundation for the rest of the material covered throughout; Part Two discusses Euclidean transformations of the plane, as well as groups and their use in studying transformations; and Part Three covers inversive and projective geometry as natural extensions of Euclidean geometry. In addition to featuring real-world applications throughout, Classical Geometry: Euclidean, Transformational, Inversive, and Projective includes: * Multiple entertaining and elegant geometry problems at the end of each section for every level of study * Fully worked examples with exercises to facilitate comprehension and retention * Unique topical coverage, such as the theorems of Ceva and Menalaus and their applications * An approach that prepares readers for the art of logical reasoning, modeling, and proofs The book is an excellent textbook for courses in introductory geometry, elementary geometry, modern geometry, and history of mathematics at the undergraduate level for mathematics majors, as well as for engineering and secondary education majors. The book is also ideal for anyone who would like to learn the various applications of elementary geometry.
(source: Nielsen Book Data)9781118679197 20180530

• Preface v PART I EUCLIDEAN GEOMETRY 1 Congruency 3 1.1 Introduction 3 1.2 Congruent Figures 6 1.3 Parallel Lines 12 1.3.1 Angles in a Triangle 13 1.3.2 Thales' Theorem 14 1.3.3 Quadrilaterals 17 1.4 More About Congruency 21 1.5 Perpendiculars and Angle Bisectors 24 1.6 Construction Problems 28 1.6.1 The Method of Loci 31 1.7 Solutions to Selected Exercises 33 1.8 Problems 38 2 Concurrency 41 2.1 Perpendicular Bisectors 41 2.2 Angle Bisectors 43 2.3 Altitudes 46 2.4 Medians 48 2.5 Construction Problems 50 2.6 Solutions to the Exercises 54 2.7 Problems 56 3 Similarity 59 3.1 Similar Triangles 59 3.2 Parallel Lines and Similarity 60 3.3 Other Conditions Implying Similarity 64 3.4 Examples 67 3.5 Construction Problems 75 3.6 The Power of a Point 82 3.7 Solutions to the Exercises 87 3.8 Problems 90 4 Theorems of Ceva and Menelaus 95 4.1 Directed Distances, Directed Ratios 95 4.2 The Theorems 97 4.3 Applications of Ceva's Theorem 99 4.4 Applications of Menelaus' Theorem 103 4.5 Proofs of the Theorems 115 4.6 Extended Versions of the Theorems 125 4.6.1 Ceva's Theorem in the Extended Plane 127 4.6.2 Menelaus' Theorem in the Extended Plane 129 4.7 Problems 131 5 Area 133 5.1 Basic Properties 133 5.1.1 Areas of Polygons 134 5.1.2 Finding the Area of Polygons 138 5.1.3 Areas of Other Shapes 139 5.2 Applications of the Basic Properties 140 5.3 Other Formulae for the Area of a Triangle 147 5.4 Solutions to the Exercises 153 5.5 Problems 153 6 Miscellaneous Topics 159 6.1 The Three Problems of Antiquity 159 6.2 Constructing Segments of Specific Lengths 161 6.3 Construction of Regular Polygons 166 6.3.1 Construction of the Regular Pentagon 168 6.3.2 Construction of Other Regular Polygons 169 6.4 Miquel's Theorem 171 6.5 Morley's Theorem 178 6.6 The Nine-Point Circle 185 6.6.1 Special Cases 188 6.7 The Steiner-Lehmus Theorem 193 6.8 The Circle of Apollonius 197 6.9 Solutions to the Exercises 200 6.10 Problems 201 PART II TRANSFORMATIONAL GEOMETRY 7 The Euclidean Transformations or Isometries 207 7.1 Rotations, Reflections, and Translations 207 7.2 Mappings and Transformations 211 7.2.1 Isometries 213 7.3 Using Rotations, Reflections, and Translations 217 7.4 Problems 227 8 The Algebra of Isometries 231 8.1 Basic Algebraic Properties 231 8.2 Groups of Isometries 236 8.2.1 Direct and Opposite Isometries 237 8.3 The Product of Reflections 241 8.4 Problems 246 9 The Product of Direct Isometries 253 9.1 Angles 253 9.2 Fixed Points 255 9.3 The Product of Two Translations 256 9.4 The Product of a Translation and a Rotation 257 9.5 The Product of Two Rotations 260 9.6 Problems 263 10 Symmetry and Groups 269 10.1 More About Groups 269 10.1.1 Cyclic and Dihedral Groups 273 10.2 Leonardo's Theorem 277 10.3 Problems 281 11 Homotheties 287 11.1 The Pantograph 287 11.2 Some Basic Properties 288 11.2.1 Circles 291 11.3 Construction Problems 293 11.4 Using Homotheties in Proofs 298 11.5 Dilatation 302 11.6 Problems 304 12 Tessellations 311 12.1 Tilings 311 12.2 Monohedral Tilings 312 12.3 Tiling with Regular Polygons 317 12.4 Platonic and Archimedean Tilings 323 12.5 Problems 330 PART III INVERSIVE AND PROJECTIVE GEOMETRIES 13 Introduction to Inversive Geometry 337 13.1 Inversion in the Euclidean Plane 337 13.2 The Effect of Inversion on Euclidean Properties 343 13.3 Orthogonal Circles 351 13.4 Compass-Only Constructions 360 13.5 Problems 369 14 Reciprocation and the Extended Plane 373 14.1 Harmonic Conjugates 373 14.2 The Projective Plane and Reciprocation 383 14.3 Conjugate Points and Lines 393 14.4 Conics 399 14.5 Problems 406 15 Cross Ratios 409 15.1 Cross Ratios 409 15.2 Applications of Cross Ratios 420 15.3 Problems 429 16 Introduction to Projective Geometry 433 16.1 Straightedge Constructions 433 16.2 Perspectivities and Projectivities 443 16.3 Line Perspectivities and Line Projectivities 448 16.4 Projective Geometry and Fixed Points 448 16.5 Projecting a Line to Infinity 451 16.6 The Apollonian Definition of a Conic 455 16.7 Problems 461 Bibliography 464 Index 469.
• (source: Nielsen Book Data)9781118679197 20180530

Features the classical themes of geometry with plentiful applications in mathematics, education, engineering, and science Accessible and reader-friendly, Classical Geometry: Euclidean, Transformational, Inversive, and Projective introduces readers to a valuable discipline that is crucial to understanding bothspatial relationships and logical reasoning. Focusing on the development of geometric intuitionwhile avoiding the axiomatic method, a problem solving approach is encouraged throughout. The book is strategically divided into three sections: Part One focuses on Euclidean geometry, which provides the foundation for the rest of the material covered throughout; Part Two discusses Euclidean transformations of the plane, as well as groups and their use in studying transformations; and Part Three covers inversive and projective geometry as natural extensions of Euclidean geometry. In addition to featuring real-world applications throughout, Classical Geometry: Euclidean, Transformational, Inversive, and Projective includes: * Multiple entertaining and elegant geometry problems at the end of each section for every level of study * Fully worked examples with exercises to facilitate comprehension and retention * Unique topical coverage, such as the theorems of Ceva and Menalaus and their applications * An approach that prepares readers for the art of logical reasoning, modeling, and proofs The book is an excellent textbook for courses in introductory geometry, elementary geometry, modern geometry, and history of mathematics at the undergraduate level for mathematics majors, as well as for engineering and secondary education majors. The book is also ideal for anyone who would like to learn the various applications of elementary geometry.
(source: Nielsen Book Data)9781118679197 20180530

• Part I: Geometry.- The Ricci flow on some generalized Wallach spaces (N.A. Abiev, A. Arvanitoyeorgos, Y.G. Nikonorov, P. Siasos).- Gaussian mean curvature flow for submanifolds in space forms (A. Borisenko, V. Rovenski).- Cantor laminations and exceptional minimal sets in codimension one foliations (G. Hector).- Integral formulas in foliations theory (K. Andrzejewski, P. Walczak, V. Rovenski).- On prescribing the mixed scalar curvature of a foliations (V. Rovenski, L. Zelenko).- The partial Ricci flow for foliations (V. Rovenski).- Osculation in general (P. Walczak).- On stability of totally geodesic unit vector fields on three-dimensional Lie groups (A. Yampolsky).- Part II: Applications.- Rotational liquid film interacted with ambient gaseous media (I. Gaissinski, Y. Levy, V. Rovenski, V. Sherbaum).- On cycles and other geometric phenomena in phase portraits of some nonlinear dynamical systems (V. Golubyatnikov, Yu. A. Gaidov).- Remez-type inequality for smooth functions (Y. Iomdin).
• (source: Nielsen Book Data)9783319046747 20160614

This volume has been divided into two parts: Geometry and Applications. The geometry portion of the book relates primarily to geometric flows, laminations, integral formulae, geometry of vector fields on Lie groups and osculation; the articles in the applications portion concern some particular problems of the theory of dynamical systems, including mathematical problems of liquid flows and a study of cycles for non-dynamical systems. This Work is based on the second international workshop entitled "Geometry and Symbolic Computations, " held on May 15-18, 2013 at the University of Haifa and is dedicated to modeling (using symbolic calculations) in differential geometry and its applications in fields such as computer science, tomography and mechanics. It is intended to create a forum for students and researchers in pure and applied geometry to promote discussion of modern state-of-the-art in geometric modeling using symbolic programs such as Maple(TM) and Mathematica(R) , as well as presentation of new results.
(source: Nielsen Book Data)9783319046747 20160614

• Preface-- Introduction: geometry and geometries-- 1. Conics-- 2. Affine geometry-- 3. Projective geometry: lines-- 4. Projective geometry: conics-- 5. Inversive geometry-- 6. Hyperbolic geometry: the disc model-- 7. Elliptic geometry: the spherical model-- 8. The Kleinian view of geometry-- Special symbols-- Further reading-- Appendix 1. A primer of group theory-- Appendix 2. A primer of vectors and vector spaces-- Appendix 3. Solutions to the problems-- Index.
• (source: Nielsen Book Data)9781139003001 20160613

This richly illustrated and clearly written undergraduate textbook captures the excitement and beauty of geometry. The approach is that of Klein in his Erlangen programme: a geometry is a space together with a set of transformations of the space. The authors explore various geometries: affine, projective, inversive, hyperbolic and elliptic. In each case they carefully explain the key results and discuss the relationships between the geometries. New features in this second edition include concise end-of-chapter summaries to aid student revision, a list of further reading and a list of special symbols. The authors have also revised many of the end-of-chapter exercises to make them more challenging and to include some interesting new results. Full solutions to the 200 problems are included in the text, while complete solutions to all of the end-of-chapter exercises are available in a new Instructors' Manual, which can be downloaded from www.cambridge.org/9781107647831.
(source: Nielsen Book Data)9781139003001 20160613

• Reality and imagination
• On problems
• Size and shape
• Time and space.

For seven years, Paul Lockhart's A Mathematician's Lament enjoyed a samizdat-style popularity in the mathematics underground, before demand prompted its 2009 publication to even wider applause and debate. An impassioned critique of K--12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living. In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science. Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can "do the math" in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.
(source: Nielsen Book Data)9780674057555 20160609