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)
Author(s): Israel M. Gelfand; Tatiana Alekseyevskaya (Gelfand)
Publisher: Birkhäuser
Year: 2020
Language: English
Pages: xxi+420
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
