This textbook is a self-contained presentation of Euclidean Geometry, a subject that has been a core part of school curriculum for centuries. The discussion is rigorous, axiom-based, written in a traditional manner, true to the Euclidean spirit. Transformations in the Euclidean plane are included as part of the axiomatics and as a tool for solving construction problems.
The textbook can be used for teaching a high school or an introductory level college course. It can be especially recommended for schools with enriched mathematical programs and for homeschoolers looking for a rigorous traditional discussion of geometry.
The text is supplied with over 1200 questions and problems, ranging from simple to challenging. The solutions sections of the book contain about 200 answers and hints to solutions and over 100 detailed solutions involving proofs and constructions. More solutions and some supplements for teachers are available in the Instructor’s Manual, which is issued as a separate book.
From the Reviews... ‘In terms of presentation, this text is more rigorous than any existing high school textbook that I know of. It is based on a system of axioms that describe incidence, postulate a notion of congruence of line segments, and assume the existence of enough rigid motions ("free mobility")…
My gut reaction to the book is, wouldn't it be wonderful if American high school students could be exposed to this serious mathematical treatment of elementary geometry, instead of all the junk that is presented to them in existing textbooks.
This book makes no concession to the TV-generation of students who want (or is it the publishers who want it for them?) pretty pictures, side bars, puzzles, games, historical references, cartoons, and all those colored images that clutter the pages of a typical modern textbook, while the mathematical content is diluted more and more with each successive edition.’
Professor Robin Hartshorne, University of California at Berkeley.
‘The textbook “Euclidean Geometry” by Mark Solomonovich fills a big gap in the plethora of mathematical textbooks – it provides an exposition of classical geometry with emphasis on logic and rigorous proofs…
I would be delighted to see this textbook used in Canadian schools in the framework of an improved geometry curriculum. Until this day comes, I highly recommend “Euclidean Geometry” by Mark Solomonovich to be used in Mathematics Enrichment Programs across Canada and the USA.’
Professor Yuly Billig, Carlton University
Author(s): Mark Solomonovich
Edition: 1
Publisher: iUniverse
Year: 2010
Language: English
Commentary: Front cover, bookmarks, OCR
Pages: 405
City: Bloomington
1. Introduction
1.1 Introductory remarks on the subject of geometry
1.2 Euclidean geometry as a deductive system. Common notions
1.3 Undefined terms. Basic definitions and first axioms
2. Mathematical Propositions
2.1 Statements; Sets of Axioms; Propositions; Theorems
2.2 Theorems: the structure and proofs
2.3 Direct, converse, inverse, and contrapositive theorems
3. Angles
3.1 Basic Notions
3.2 Kinds of angles. Perpendiculars
3.3 Vertical angles
3.4 Central angles and their corresponding arcs. Measurement of angles
4. Triangles
4.1 Broken line; polygons
4.2 Triangles; the basic notions
4.3 Axial Symmetry
4.4 Congruence of triangles. Some properties of an isosceles triangle
4.5 Exterior angle of a triangle
4.6 Relations between sides and angles in a triangle
4.7 Comparative length of the straight segment and a broken line connecting the same pair of points
4.8 Comparison of a perpendicular and the obliques
4.9 Congruence of right triangles
4.10 The perpendicular bisector of a segment and the bisector of an angle as loci
5. Construction Problems
5.1 The axioms and tools allowing geometric constructions
5.2 Basic construction problems
a) To construct an angle equal to a given angle, with the vertex at a given point and one side lying in a given line
b) To construct a triangle having given the three sides (SSS)
c) To construct a triangle having given two sides and the included angle (SAS)
d) To construct a triangle having given two angles and the included side (ASA)
e) To construct a triangle having given two sides and an angle opposite to one of them (SSA)
f) To bisect an angle (draw the symmetry axis of an angle)
g) To erect a perpendicular to a given line from a given point in the line
h) To drop a perpendicular onto a line from a given external point
5.3 Strategy for construction: analysis, construction, synthesis, investigation
6. Parallel Lines
6.1 Definition. Theorem of existence. .
6.2 Angles formed by a pair of lines and their transversal. Three tests for the parallelism. Parallel postulate (Playfair’s formulation)
6.3 Tests for the nonparallelism of two lines. Euclid’s formulation of Parallel Postulate
6.4 Angles with respectively parallel sides. Angles with respectively perpendicular sides.
6.5 The sum of the interior angles of a triangle. The sum of the interior angles of a polygon. The
sum of the exterior angles of a convex polygon
6.6 Central Symmetry
7. PARALLELOGRAM AND TRAPEZOID
7.1 Parallelogram
7.2 Particular cases of parallelograms: rectangle, rhombus (diamond), square. Symmetry properties of parallelograms
7.3 Some theorems based on properties of parallelograms
7.4 Trapezoid
7.5. a) Translations
b) Translations and symmetry in construction problems
8. CIRCLES
8.1 Shape and location
8.2 Relations between arcs and chords and their distances from the centre
8.3 Respective position of a straight line and a circle
8.4 Respective position of two circles
8.5 Inscribed angles, some other angles related to circles. Construction of tangents
8.6 Constructions using loci
8.7 Inscribed and circumscribed polygons
8.8 Four remarkable points of a triangle
9. SIMILARITY
9.1 The notion of measurement
a) The setting of the problem
b) Common measure
c) Finding the greatest common measure
d) Commensurable and incommensurable segments
e) Measurement of segments. Rational and irrational numbers
f) The numerical length of a segment. The ratio of two segments
9.2 Similarity of Triangles
a) Preliminary remarks
b) Corresponding sides; definition and existence of similar triangles
9.3 Three tests for the similarity of triangles
9.4 Similarity of right triangles
9.5 Similarity of polygons
9.6 Similarity of general plane figures
a) Construction of a figure similar to a given one
b) Similarity of circles
9.7 Similarity method in construction problems
9.8 Some theorems on proportional segments
a) Segments related to parallel lines
b) Properties of bisectors
9.9 Metric relations in triangles and some other figures
a) Mean proportionals in right triangles
b) The Pythagorean Theorem
9.10 Proportional segments in circles
9.11 Trigonometric functions of acute angle
a) Definitions
b) Basic trigonometric identities
c) Construction of angles given their trigonometric functions
d) Behaviour of trigonometric functions of acute angles
e) Solving triangles using trigonometry
9.12 Applications of algebra in geometry
a) Construction of the golden ratio
b) General principles of algebraic methods for solving geometric problems
c) The construction of some elementary formulae
10. REGULAR POLYGONS AND CIRCUMFERENCE
10.1 Regular polygons
10.2 The circumference of a circle
10.3 The circumference as a limit
11. AREAS
11.1 Areas of polygons
a) Basic suggestions (principles)
b) Measurement of area
c) Area of a rectangle
d) Areas of polygons
e) Pythagorean Theorem in terms of areas
11.2 Areas of similar figures
11.3 The area of a circle
12. ANSWERS, HINTS (1)
SOLUTIONS, HINTS (2)
APPENDIX: Why this Textbook is Written the Way It Is
