1.1. Some constructions 5 Figure 1.1.6 There is a point we might take a moment to emphasize, even though we cannot make this with the emphasis that is appropriate until we formulate the trisection problem in precise terms. Namely, in the above constructions phrases such as “pick a point” or “choose a radius” are often used. In the context of the trisection problem we are constrained as to which points and radii can be chosen. Indeed, this is at the heart of the problem. At this juncture the reader will have to have a bit of faith that in these construc- tions the points and radii that are required to complete the constructions can be chosen under the constraints that will be imposed by the trisection framework. In fact, even a casual reading of the proofs shows that there is a great deal of latitude in picking the points and radii, so this seemingly random choice can be adjusted and will cause no problems under the con- straints imposed by the trisection problem. Of course later, after learning the constraints, the reader can return to these constructions and verify that all can be carried out in a manner consistent with the restrictions we must live with. The constructions above are basic to what we will do and will be used repeatedly in this chapter with no specific reference to one of the preceding propositions. Exercises 1. Let ΔABC be an isosceles triangle with |AB| = |AC|. From A draw a line perpendicular to BC and let D be the point where this line meets BC. Show that |CD| = |BD|. See Figure 1.1.7. 2. Show that any line segment can be bisected using a compass and straight- edge alone. 3. Let ∠XY Z be a given angle and let OA be a given line segment. Show that using compass and straightedge alone you can construct a line seg- ment OB such that ∠AOB = ∠XY Z. See Figure 1.1.8. 4. Given a polygonal path ABCDE, show that it is possible to construct a line segment having length |AB|+|BC|+|CD|+|DE| using straightedge and compass alone.
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