6 - Circle Geometry

Summary

Introduction

This chapter continues the subject of geometry in the plane. There are many types of problems that use circles in their solution, some involving triangles as well as circles. Many of the problems that involve circles are most easily solved using equations to represent that circle, but these will be postponed to a later chapter. Here we consider only those problems that strictly involve plane geometry.

There are numerous definitions and results in this material, and it is important to have complete familiarity with the notation.

Definitions

We begin with the basic definitions and include here all the terminology that will be used for the problems that involve circles. The most basic and frequently used are those involving the area and circumference of a circle.

Definition 1 Circles:

1. • A circle is a set of all points that are a fixed distance from a given point.

2. • The center of the circle is the given point.

3. • Any line segment from the center that has the fixed distance as its length is a radius of the circle.

The term radius is also used to describe the fixed distance from the center to the points on the circle. This could cause confusion, but context will make the distinction clear.

Definition 2 Lines and Circles:

1. • A line that has exactly one point in common with a circle is called a tangent line to the circle.

2. • A line that intersects two points of a circle is called a secant line of the circle. The line segment of the secant line that joins the two points on the circle is called a chord.

3. • A diameter of a circle is a chord that passes through the center of the circle.

The term diameter is also used to describe the length of a diameter, which is twice the length of the radius.

A basic way to relate circle geometry to triangle geometry is to use an angle that has its vertex at the center of the circle.

DEFINITION 3 A central angle of a circle is an angle whose vertex is at the center of the circle. A central angle partitions the circle into two portions. The larger and smaller portions are the major arc and minor arc, Respectively.