28 1. Trisecting Angles 1.9. Marks on the straightedge Most things you read on ruler and compass constructions do not allow the straightedge to have marks on it. My suspicion is that this was the way the ancient Greeks phrased it, but I am uncertain. The approach taken here is to present a mathematical problem for solution, and in that context the presence or absence of marks on the straightedge is irrelevant. In fact, as we approached the problem, we never talked about marks on the straightedge. On the other hand, we are given the two starting points at the origin and (1, 0), so why can’t we just take those as marks on the straightedge? If you like, why not notch our straightedge to indicate the points O and I? In fact, that is what we do in this section. We assume we are given a straightedge with marks denoting the unit distance, and we present the usual “proof” that any angle can be trisected. This, however, does not solve the trisection problem as we have stated it. The reason for this does not lie in the fact that the straightedge has marks on it but is somewhat more subtle. We will perform the construction and ask the reader to stay alert and try to spot where the diﬃculty arises. After the construction we’ll discuss it and point out where the argument goes awry. Assume there are two marks on the straightedge that are 1 unit apart and let an angle be given with θ radians. Assume that the vertex of the angle is at the origin O and construct a circle with center O and radius 1. Let A and B = (1, 0) be the points on the sides of the angle where they intersect this circle. See Figure 1.9.1. Figure 1.9.1 The reader might want to refer to Figure 1.9.2 while reading this para- graph. Place the straightedge so that it goes through the point A and forms a secant line for the circle. Move the straightedge around, always forming a secant, until one of the two marks lies on the circle and the other on the x-axis. This requires some physical dexterity but can be done. In fact, if the straightedge hits the point A and one mark is at (−1, 0) with the other mark lying below the x-axis, begin to raise the straightedge so that the first mark lies on the circle and the straightedge continues to hit the point A, at a different place on the straightedge. While doing this the second mark

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