# 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

**. 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**

*straightedge**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

**. 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**

*modern compass***which collapses to zero radius when removed from the page.**

*classical compass*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.**

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

*Squaring the circle*Get *Classical Geometry: Euclidean, Transformational, Inversive, and Projective* now with the O’Reilly learning platform.

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