Special Cases of Apollonius' Problem - Historical Introduction

Historical Introduction

Like most branches of mathematics, Euclidean geometry is concerned with proofs of general truths from a minimum of postulates. For example, a simple proof would show that at least two angles of an isosceles triangle are equal. One important type of proof in Euclidean geometry is to show that a geometrical object can be constructed with a compass and an unmarked straightedge; an object can be constructed if and only if (iff) (something about no higher than square roots are taken). Therefore, it is important to determine whether an object can be constructed with compass and straightedge and, if so, how it may be constructed.

Euclid developed numerous constructions with compass and straightedge. Examples include: regular polygons such as the pentagon and hexagon, a line parallel to another that passes through a given point, etc. Many rose windows in Gothic Cathedrals, as well as some Celtic knots, can be designed using only Euclidean constructions. However, some geometrical constructions are not possible with those tools, including the heptagon and trisecting an angle.

Apollonius contributed many constructions, namely, finding the circles that are tangent to three geometrical elements simultaneously, where the "elements" may be a point, line or circle.