Geometry

This text is the fifth and final in the series of educational books written by Israel Gelfand with his colleagues for high school students. These books cover the basics of mathematics in a clear and simple format – the style Gelfand was known for internationally. Gelfand prepared these materials so as to be suitable for independent studies, thus allowing students to learn and practice the material at their own pace without a class. Geometry takes a different approach to presenting basic geometry for high-school students and others new to the subject. Rather than following the traditional axiomatic method that emphasizes formulae and logical deduction, it focuses on geometric constructions. Illustrations and problems are abundant throughout, and readers are encouraged to draw figures and “move” them in the plane, allowing them to develop and enhance their geometrical vision, imagination, and creativity. Chapters are structured so that only certain operations and the instruments to perform these operations are available for drawing objects and figures on the plane. This structure corresponds to presenting, sequentially, projective, affine, symplectic, and Euclidean geometries, all the while ensuring students have the necessary tools to follow along. Geometry is suitable for a large audience, which includes not only high school geometry students, but also teachers and anyone else interested in improving their geometrical vision and intuition, skills useful in many professions. Similarly, experienced mathematicians can appreciate the book’s unique way of presenting plane geometry in a simple form while adhering to its depth and rigor. “Gelfand was a great mathematician and also a great teacher. The book provides an atypical view of geometry. Gelfand gets to the intuitive core of geometry, to the phenomena of shapes and how they move in the plane, leading us to a better understanding of what coordinate geometry and axiomatic geometry seek to describe.” - Mark Saul, PhD, Executive Director, Julia Robinson Mathematics Festival “The subject matter is presented as intuitive, interesting and fun. No previous knowledge of the subject is required. Starting from the simplest concepts and by inculcating in the reader the use of visualization skills, [and] after reading the explanations and working through the examples, you will be able to confidently tackle the interesting problems posed. I highly recommend the book to any person interested in this fascinating branch of mathematics.” - Ricardo Gorrin, a student of the Extended Gelfand Correspondence Program in Mathematics (EGCPM) Contents......Page 5 for the series of books written by Israel Gelfand for high-school students......Page 8 What is special about this book? Why and for whom was it written?......Page 10 About the process of writing Geometry.......Page 11 Acknowledgements.......Page 14 Geometry is the simplest model of spatial relationships in our world......Page 15 Structure of this book and how to read it......Page 17 1.1 What is a point and what is a line?......Page 20 1.2 Operations available in Chapter I......Page 22 1.3 Ray, segment, half-plane......Page 23 1.4 Constructions with a straightedge......Page 25 2.1 Notion of an angle......Page 29 2.2 Some types of angles......Page 32 3.1 Configurations of three lines......Page 35 3.2 Triangles......Page 40 4 Four lines. Quadrilaterals......Page 44 6 Projection from a point onto a line......Page 48 7 Dual configurations in projective geometry......Page 59 8 Desargues configuration......Page 64 9 Dual Desargues configuration......Page 69 10 Algebraic notation or “computer presentation” of configurations......Page 75 11 Polygons and n straight lines......Page 78 12 Convex polygons, convex hull of n points......Page 81 13 Solution of Exercise 3 with the help of a Desargues configuration......Page 84 14 Overview of Chapter I......Page 89 1 Parallel straight lines......Page 90 2 Operations available in Chapter II......Page 92 3.1 Transitivity of parallel lines......Page 93 3.3 Reflexivity of parallel lines......Page 94 4.1 Equality of segments lying on parallel lines......Page 95 4.2 Construction of equal segments on parallel lines......Page 98 Properties of equal segments lying on parallel lines......Page 100 4.3 Construction of a segment of double length......Page 102 4.4 Division of a segment into equal parts......Page 103 5.1 Definition of a parallelogram......Page 106 5.2 Properties of parallelograms......Page 107 5.3 Proof of the Lemma......Page 112 5.4 More properties of parallelograms......Page 115 6.1 Bimedian of a triangle......Page 118 6.2 Median of a triangle......Page 124 7 Trapezoids......Page 126 8 The Minkowsky addition of two figures......Page 132 9 Parallel projection......Page 134 10.1 Parallel translation of a figure......Page 139 Sum of the exterior angles of a polygon......Page 145 Defining the same parallel translation by indicating different pairs of points......Page 147 10.3 Parallel translation on a line......Page 149 11 Central symmetry on the plane......Page 151 11.1 Sequences of parallel translations and central symmetries. The relation between central symmetry and parallel translation......Page 156 12.1 Vectors and parallel translations......Page 164 12.2 Addition of vectors......Page 165 12.3 Vectors lying on parallel lines......Page 170 12.4 Subtraction of vectors......Page 173 12.5 More problems on vectors......Page 175 13 Overview of Chapter II......Page 178 1 Why we cannot define equal segments in Chapter II......Page 180 2.1 Variation of the Desargues configuration in the case of parallel lines......Page 183 2.3 A property of parallel translation......Page 186 3.1 Addition and subtraction......Page 189 3.2 Multiplication and division......Page 192 4.1 Number axis......Page 196 4.2 Finding the coordinate of a point and length of a segment......Page 199 5 Affine coordinate systems on the plane......Page 200 1 The area of a figure......Page 208 2.1 Constructing parallelograms with rational area......Page 213 Changing the length of the sides of a unit parallelogram......Page 218 Changing the direction of the sides of a unit parallelogram......Page 220 2.3 How to measure the area of a parallelogram......Page 223 3 Area of a triangle......Page 224 4 Area of a trapezoid......Page 234 5 Area of a polygon......Page 236 6 More problems on areas......Page 243 7 How to measure the area of a figure......Page 246 8 Overview of Chapter III......Page 249 1 Operations available in Chapter IV......Page 250 1.1 Properties of a circle. Some related definitions......Page 252 2 Comparing segments......Page 254 3.1 Comparing angles. Degree measure......Page 256 Arc degree measure......Page 257 3.3 Addition of angles......Page 260 3.4 Vertical angles and angles with respectively parallel sides......Page 267 4.1 Turns and reflections......Page 271 4.2 Consecutive operations with a figure. Congruent figures......Page 273 5 Elements of a triangle. Congruent triangles......Page 276 6 Construction of a triangle from its elements......Page 278 Additional constructions of a triangle from its elements......Page 280 7.1 Relations between the sides of a triangle......Page 282 7.2 Relations between the angles of a triangle......Page 285 7.3 More about angles in a triangle......Page 287 8 Properties of a triangle. Particular kinds of triangles......Page 290 8.1 The isosceles triangle......Page 292 8.2 Equilateral triangle......Page 295 8.3 Right triangle......Page 296 9.1 Measurement of area. Area of a rectangle......Page 300 9.2 Area of a triangle......Page 301 10.1 The Pythagorean theorem......Page 304 10.2 The use of the Pythagorean theorem in arbitrary triangles......Page 309 10.3 Heron’s formula for the area of a triangle......Page 313 11.1 Perpendicular from a point to a line......Page 315 11.2 Distance from a point to a line......Page 317 11.3 The locus of points lying at equal distance from two given points......Page 318 11.4 The locus of points lying at equal distance from two given lines. Two definitions of an angle bisector......Page 320 11.5 Angles with respectively perpendicular sides......Page 326 12.2 The angle bisector......Page 327 12.3 The perpendicular bisector......Page 331 12.4 The altitudes......Page 332 12.5 Special lines of a triangle at a glance......Page 334 12.6 Special points in a triangle......Page 336 13.1 Definitions of special quadrilaterals......Page 337 13.2 Regular polygons......Page 339 13.3 The sum of the angles of a polygon......Page 340 14.1 Trapezoid......Page 343 Area of a trapezoid......Page 344 14.2 Parallelogram......Page 345 Area of a parallelogram......Page 346 14.3 Rectangle......Page 347 Area of a rhombus......Page 348 Area of a square......Page 351 15 Similarity......Page 352 15.1 Similar triangles......Page 353 15.2 Similarity of polygons and area of similar polygons......Page 356 15.3 A third proof of the Pythagorean theorem......Page 357 16.1 Circles passing through a point......Page 358 16.2 Circles passing through two points......Page 359 16.3 Circles passing through three points......Page 361 17.1 The relative positions of a circle and a line......Page 362 Circles tangent to one straight line......Page 363 Circles tangent to three straight lines......Page 365 18.1 The relative positions of two circles......Page 366 18.2 The relative positions of three circles......Page 370 19 Circles and angles......Page 372 19.1 Inscribed angles......Page 373 19.3 An angle with its vertex outside a circle......Page 379 Extreme positions of a circle and an angle......Page 381 19.4 An angle which a segment subtends......Page 384 20.1 Inscribed and circumscribed triangles......Page 390 20.2 Some exercises on inscribed and circumscribed triangles......Page 395 20.3 The area of a circumscribed triangle. The area of an inscribed triangle......Page 397 21.1 Inscribed polygons......Page 400 21.2 Inscribed quadrilaterals. Ptolemy’s theorem......Page 402 21.3 Some problems on inscribed quadrilaterals......Page 404 21.4 The relation between a circle and a regular polygon with n vertices......Page 407 22.1 Circumference......Page 410 22.2 The number π......Page 412 22.3 Length of an arc......Page 415 22.4 Radian measure of an angle......Page 416 23.1 Area of a regular polygon......Page 417 23.3 Area of a sector......Page 418 24 Overview of Chapter IV......Page 419 Glossary......Page 421 This text is the fifth and final in the series of educational books written by Israel Gelfand with his colleagues for high school students. These books cover the basics of mathematics in a clear and simple format - the style Gelfand was known for internationally. Gelfand prepared these materials so as to be suitable for independent studies, thus allowing students to learn and practice the material at their own pace without a class. Geometry takes a different approach to presenting basic geometry for high-school students and others new to the subject. Rather than following the traditional axiomatic method that emphasizes formulae and logical deduction, it focuses on geometric constructions. Illustrations and problems are abundant throughout, and readers are encouraged to draw figures and "move" them in the plane, allowing them to develop and enhance their geometrical vision, imagination, and creativity. Chapters are structured so that only certain operations and the instruments to perform these operations are available for drawing objects and figures on the plane. This structure corresponds to presenting, sequentially, projective, affine, symplectic, and Euclidean geometries, all the while ensuring students have the necessary tools to follow along. Geometry is suitable for a large audience, which includes not only high school geometry students, but also teachers and anyone else interested in improving their geometrical vision and intuition, skills useful in many professions. Similarly, experienced mathematicians can appreciate the books unique way of presenting plane geometry in a simple form while adhering to its depth and rigor. "Gelfand was a great mathematician and also a great teacher. The book provides an atypical view of geometry. Gelfand gets to the intuitive core of geometry, to the phenomena of shapes and how they move in the plane, leading us to a better understanding of what coordinate geometry and axiomatic geometry seek to describe." Mark S aul, PhD, Executive Director, Julia Robinson Mathematics Festival "The subject matter is presented as intuitive, interesting and fun. No previous knowledge of the subject is required. Starting from the simplest concepts and by inculcating in the reader the use of visualization skills, [and] after reading the explanations and working through the examples, you will be able to confidently tackle the interesting problems posed. I highly recommend the book to any person interested in this fascinating branch of mathematics." Ricardo Gorrin, a student of the Extended Gelfand Correspondence Program in Mathematics (EGCPM)