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Geometric Constructions

Ku, Yin Bon (Albert), SAVAŞ ORHAN, Mar 10, 2018

This is an interactive course on geometric constructions, a fascinating topic that has been ignored by the mainstream mathematics education. It is all about drawing geometric figures using specific drawing tools like straightedge, compass and so on. This classical topic in geometry is important because

the foundation of geometry is mostly inspired by what we can do with all these drawing tools, and
it involves a lot of beautiful mathematics that shows the interplay between geometry and algebra.

There are numerous construction exercises in this course. When you successfully construct the required geometric figure(s) in each exercise, the "Well done!" message will appear. Prerequisite for this course:

A little bit set theory
High school geometry and algebra

(Note: This GeoGebraBook is an outgrowth of an undergraduate math course that I offer in the Hong Kong University of Science and Technology)

Euclidean Constructions
1. Introduction
2. Copy a Line Segment
3. Bisect an Angle
4. Construct an Equilateral Triangle
5. Copy a Triangle
6. Copy an Angle
7. Construct a Perpendicular Bisector and Midpoint
8. Construct a Perpendicular Line
9. Construct a Parallel Line
10. Construct a Point of Division
11. Circumcircle of a Triangle
12. Using Euclidean Compass
13. Tangents of a Circle
14. Inscribe a Circle in a Triangle
Constructibility
1. Introduction
2. Addition and Subtraction
3. Division
4. Multiplication
5. Squaring a Rectangle
6. Taking Square Root
7. Constructible Points and Numbers
8. Field
9. Field Extensions
Three Classical Problems
1. Introduction
2. The Main Theorem
3. Doubling a Cube
4. Double Mean Proportionals
5. The Solution of Menaechmus
6. The Solution of Erathostenes
7. Trisecting an Angle
8. Which angle can't be trisected?
9. Archimedes' Method of Angle Trisection
10. Squaring a Circle
11. Theorem of Lindemann
12. Quadratrix of Hippias
Compass-only Constructions
Straightedge-only Constructions
Regular Polygons

Information

1.

Euclidean Constructions

Euclidean constructions - Drawing geometric figures using straightedge and compass

1. Introduction
2. Copy a Line Segment
3. Bisect an Angle
4. Construct an Equilateral Triangle
5. Copy a Triangle
6. Copy an Angle
7. Construct a Perpendicular Bisector and Midpoint
8. Construct a Perpendicular Line
9. Construct a Parallel Line
10. Construct a Point of Division
11. Circumcircle of a Triangle
12. Using Euclidean Compass
13. Tangents of a Circle
14. Inscribe a Circle in a Triangle

1.1.

Introduction

Straightedge

Straightedge can be regarded as a ruler without any marking. The following are exactly what a straightedge can do:

Draw a unique straight line through two distinct points
Extend a line segment arbitrarily far in both direction

Notice that it cannot be used for measuring length.

Compass

Compass is a V-shaped drawing tool with a sharp point on one arm and a pencil on another arm. The two arms are joined by a hinge so that the openings of the arms is adjustible. Obviously, a compass is mainly used for constructing a circle centered at any given point with any given radius. There are two types of compasses:

Modern compass - We can keep the opening fixed when the compass leaves the plane and carry to another location for construction.
Euclidean (or collapsible) compass - The compass "forgets" the width of the opening when the compass leaves the plane i.e. we cannot keep the opening fixed.

Euclidean Constructions

Euclid of Alexandria (around 300 BC) was a Greek mathematician, who wrote the monumental 13-volume treatise on geometry called "The Elements (幾何原本)". It laid the foundation of geometry and has been widely considered as the most influential textbook ever written. All geometric constructions in "The Elements" only involves straightedge and Euclidean compass. Therefore, such constructions are usually called Euclidean constructions. There are quite a number of important propositions in "The Elements" which are actually Euclidean constructions. Let us list a few of them here:

Proposition 1 in Book I - Construct an equilateral triangle having a given segment as one side.
Proposition 9 in Book I - Construct the angle bisector of a given angle.
Proposition 1 in Book III - Given three non-collinear points, construct the center of the circle containing the three points.
Proposition 11 in Book IV - Inscribe a regular pentagon (5-sided polygon) in a given circle.

In this chapter, you will be guided to do various constructions using straightedge and compass. For simplicity, you are allowed to use modern compass instead of the more restrictive Euclidean compass. A natural question to think about: Is a modern compass really more powerful than a Euclidean compass? In other words, does there exist something that can be constructed by straightedge and modern compass but not by straightedge and Euclidean compass? We will answer this question later in this chapter.

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Constructibility

2.1.

Introduction

In this chapter, we are going to investigate the following fundamental problem: What exactly can be constructed by straightedge and compass only?
Very generally speaking, we attack this problem by showing that the set of lengths (of line segments) that can be constructed by straightedge and compass only and their negative counterparts, which is usually called the set of constructible numbers, possesses a very nice algebraic structure so that we can obtain an elegant criterion to check whether a real number belongs to the above set.

Any positive real number can be expressed as the length of a line segment. Therefore, it makes sense to do arithmetic in a geometric way. We will briefly go through how addition, subtraction, multiplication, division and taking square root can be done by Euclidean constructions. Also, by a combination of these operations, we can produce many more line segments of different lengths in terms of the given ones.

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Three Classical Problems

1. Introduction
2. The Main Theorem
3. Doubling a Cube
4. Double Mean Proportionals
5. The Solution of Menaechmus
6. The Solution of Erathostenes
7. Trisecting an Angle
8. Which angle can't be trisected?
9. Archimedes' Method of Angle Trisection
10. Squaring a Circle
11. Theorem of Lindemann
12. Quadratrix of Hippias

3.1.

Introduction

After learning knowledge about constructible numbers in the last chapter, we are now ready to tackle the so-called "three classical problems" in ancient Greek geometry. They are

Doubling a cube
Trisecting an angle
Squaring a circle

These problems were extremely influential in the development of Greek geometry. It turns out that none of the above can be done by Euclidean constructions. We will use the "main theorem" that will be introduced in the next page to prove that it is impossible to double a cube or trisect an angle by straightedge and compass only. As for squaring a circle, a more advanced theorem will be needed.

Compass-only Constructions

Coming soon!

This chapter does not contain any resources yet.

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Straightedge-only Constructions

Coming soon!

This chapter does not contain any resources yet.

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Regular Polygons

Coming soon!

This chapter does not contain any resources yet.

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Geometric Constructions

Table of Contents

Information

Euclidean Constructions

Introduction

Straightedge

Compass

Euclidean Constructions

Constructibility

Introduction

Three Classical Problems

Introduction

Compass-only Constructions

Straightedge-only Constructions

Regular Polygons