2 1. Trisecting Angles this is that it is a beautiful piece of mathematics and its position in the development of mathematics is critical, not just curious. The problem is a purely geometric one the solution, however, is algebraic. The application of algebra to geometry was one of the great advances in mathematics, and the solution of the trisection problem is one of the best examples of this technique at work a prime example of a mathematical connection. Some historical notes are contained in §1.10. 1.1. Some constructions We start with two tools: a compass and a straightedge. The straightedge has no marks on it indicating distance. But in a certain sense we will put marks on the straightedge, so there will be nothing amiss in assuming you are given a standard ruler. This is a subtle point of contention and misunderstanding. Later in §1.9 we will present a construction using a straightedge with marks that trisects any angle. In our scheme, however, this does not solve the problem as there is a step in the construction that is illegitimate. But all this comes later, so for the time being assume you have a simple straightedge as opposed to a ruler. The compass has no markings on it either, needless to say. Before going further, it is perhaps good to review from basic geom- etry some constructions possible with a straightedge and compass. The reader should remain vigilant that only a straightedge and compass are used. (There will be some who might think a review of such elementary facts from plane geometry is inappropriate in a text meant for seniors in college. In fact my experience says otherwise. In addition, when we precisely formulate the problem, it would be beneficial for the reader to be able to look at these constructions and see that all remains within the constraints imposed on the solution of the problem.) If A and B are two points in the plane, then |AB| denotes the length of the line segment from A to B. 1.1.1. Proposition. If any angle is given, using straightedge and compass alone it is possible to bisect the angle. Figure 1.1.1 Proof. Let ∠AOB be the given angle as in Figure 1.1.1. Choose any radius r for the compass and, with O as center, construct the arc of the circle of
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