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 Preface for the series of books written by Israel Gelfand for high-school students Preface What is special about this book? Why and for whom was it written? About the process of writing Geometry. Acknowledgements. Introduction Geometry is the simplest model of spatial relationships in our world Structure of this book and how to read it Chapter I Points and Lines: A Look at Projective Geometry 1 Points and lines 1.1 What is a point and what is a line? 1.2 Operations available in Chapter I 1.3 Ray, segment, half-plane 1.4 Constructions with a straightedge 2 Two lines and an angle 2.1 Notion of an angle 2.2 Some types of angles 3 Three lines 3.1 Configurations of three lines 3.2 Triangles 4 Four lines. Quadrilaterals 5 Five lines 6 Projection from a point onto a line 7 Dual configurations in projective geometry 8 Desargues configuration 9 Dual Desargues configuration 10 Algebraic notation or “computer presentation” of configurations 11 Polygons and n straight lines 12 Convex polygons, convex hull of n points 13 Solution of Exercise 3 with the help of a Desargues configuration 14 Overview of Chapter I Chapter II Parallel Lines: A Look at Affine Geometry PART I. Lines and segments 1 Parallel straight lines 2 Operations available in Chapter II 3 Properties of parallel lines 3.1 Transitivity of parallel lines 3.2 Symmetry of parallel lines 3.3 Reflexivity of parallel lines 4 Segments lying on parallel lines 4.1 Equality of segments lying on parallel lines 4.2 Construction of equal segments on parallel lines Properties of equal segments lying on parallel lines 4.3 Construction of a segment of double length 4.4 Division of a segment into equal parts PART II. Figures 5 Parallelograms 5.1 Definition of a parallelogram 5.2 Properties of parallelograms 5.3 Proof of the Lemma 5.4 More properties of parallelograms 6 Triangles 6.1 Bimedian of a triangle 6.2 Median of a triangle 7 Trapezoids PART III. Operations with figures 8 The Minkowsky addition of two figures 9 Parallel projection 10 Parallel translation 10.1 Parallel translation of a figure 10.2 Translation of the plane Sum of the exterior angles of a polygon Defining the same parallel translation by indicating different pairs of points 10.3 Parallel translation on a line 11 Central symmetry on the plane 11.1 Sequences of parallel translations and central symmetries. The relation between central symmetry and parallel translation 12 Vectors 12.1 Vectors and parallel translations 12.2 Addition of vectors 12.3 Vectors lying on parallel lines 12.4 Subtraction of vectors 12.5 More problems on vectors 13 Overview of Chapter II Appendix for Chapter II 1 Why we cannot define equal segments in Chapter II 2 Parallel lines, equal segments, and the Desargues configuration 2.1 Variation of the Desargues configuration in the case of parallel lines 2.2 Transitivity of equal segments 2.3 A property of parallel translation 3 Arithmetic operations with segments 3.1 Addition and subtraction 3.2 Multiplication and division 4 Segments and rational numbers 4.1 Number axis 4.2 Finding the coordinate of a point and length of a segment 5 Affine coordinate systems on the plane Chapter III Area: A Look at Symplectic Geometry 1 The area of a figure 2 Area of a parallelogram 2.1 Constructing parallelograms with rational area 2.2 Different unit parallelograms Changing the length of the sides of a unit parallelogram Changing the direction of the sides of a unit parallelogram 2.3 How to measure the area of a parallelogram 2.4 How a diagonal of a parallelogram divides its area 3 Area of a triangle 4 Area of a trapezoid 5 Area of a polygon 6 More problems on areas 7 How to measure the area of a figure 8 Overview of Chapter III Chapter IV Circles: A Look at Euclidean Geometry PART I. Introduction to the circle 1 Operations available in Chapter IV 1.1 Properties of a circle. Some related definitions 2 Comparing segments 3 Angles 3.1 Comparing angles. Degree measure Arc degree measure 3.2 Construction of equal angles 3.3 Addition of angles 3.4 Vertical angles and angles with respectively parallel sides 4 Operations with figures 4.1 Turns and reflections 4.2 Consecutive operations with a figure. Congruent figures PART II. The geometry of the triangle and other figures 5 Elements of a triangle. Congruent triangles 6 Construction of a triangle from its elements Additional constructions of a triangle from its elements 7 Relations between elements of a triangle 7.1 Relations between the sides of a triangle 7.2 Relations between the angles of a triangle 7.3 More about angles in a triangle 8 Properties of a triangle. Particular kinds of triangles 8.1 The isosceles triangle 8.2 Equilateral triangle 8.3 Right triangle 9 Area in Euclidean geometry 9.1 Measurement of area. Area of a rectangle 9.2 Area of a triangle 10 The Pythagorean theorem and its applications 10.1 The Pythagorean theorem 10.2 The use of the Pythagorean theorem in arbitrary triangles 10.3 Heron’s formula for the area of a triangle 11 Relations between lines and points 11.1 Perpendicular from a point to a line 11.2 Distance from a point to a line 11.3 The locus of points lying at equal distance from two given points 11.4 The locus of points lying at equal distance from two given lines. Two definitions of an angle bisector 11.5 Angles with respectively perpendicular sides 12 Special lines and special points in a triangle 12.1 The median 12.2 The angle bisector 12.3 The perpendicular bisector 12.4 The altitudes 12.5 Special lines of a triangle at a glance 12.6 Special points in a triangle 13 Polygons 13.1 Definitions of special quadrilaterals 13.2 Regular polygons 13.3 The sum of the angles of a polygon 14 Summary of facts about different quadrilaterals 14.1 Trapezoid Area of a trapezoid 14.2 Parallelogram Area of a parallelogram 14.3 Rectangle Area of a rectangle 14.4 Rhombus Area of a rhombus 14.5 Square Area of a square 15 Similarity 15.1 Similar triangles 15.2 Similarity of polygons and area of similar polygons 15.3 A third proof of the Pythagorean theorem PART III. Circles 16 Circles and points 16.1 Circles passing through a point 16.2 Circles passing through two points 16.3 Circles passing through three points 17 Circles and lines 17.1 The relative positions of a circle and a line 17.2 Circles tangent to one, two and three straight lines Circles tangent to one straight line Circles tangent to three straight lines 18 Two or more circles 18.1 The relative positions of two circles 18.2 The relative positions of three circles 19 Circles and angles 19.1 Inscribed angles 19.2 An angle with its vertex inside a circle 19.3 An angle with its vertex outside a circle Extreme positions of a circle and an angle 19.4 An angle which a segment subtends 20 A circle and a triangle 20.1 Inscribed and circumscribed triangles 20.2 Some exercises on inscribed and circumscribed triangles 20.3 The area of a circumscribed triangle. The area of an inscribed triangle 21 Circles and polygons 21.1 Inscribed polygons 21.2 Inscribed quadrilaterals. Ptolemy’s theorem 21.3 Some problems on inscribed quadrilaterals 21.4 The relation between a circle and a regular polygon with n vertices 22 Circumference and arc 22.1 Circumference 22.2 The number π 22.3 Length of an arc 22.4 Radian measure of an angle 23 Disks and sectors 23.1 Area of a regular polygon 23.2 Area of a disk 23.3 Area of a sector 24 Overview of Chapter IV Glossary 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)