Classical Geometry: Euclidean, Transformational, Inversive, and Projective

CHAPTER 6

MISCELLANEOUS TOPICS

6.1 The Three Problems of Antiquity

In some of the earlier chapters we had sections on construction problems. In this chapter, we expand further and describe some useful techniques.

The object is to draw geometric figures in the plane using two simple tools:

1. A straightedge. This is a device for drawing a straight line through any two given points. It is of arbitrary length—that is, you can draw a line as long as you need that passes through the given points. Note that this means that given a segment AB you can extend this segment: take two points on the segment and draw the line through those two points. Note also that a straightedge is not a ruler. You cannot use a straightedge to measure distances. A ruler is a different tool.

2. A modern compass. This is a device for drawing arcs and circles given any point as center and the length of any given segment as radius. The modern compass holds its radius when it is lifted from the page, as opposed to the classical compass which collapses to zero radius when removed from the page.

The constructions that can be accomplished using these two basic tools are called Euclidean constructions.

There are certain famous construction problems that have been shown to be impossible in the sense that they cannot be accomplished with a straightedge and compass. These are:

1. Squaring the circle. The problem is to construct a square of the same area as a given circle.

Squaring the circle amounts to the following: ...