8 1. Trisecting Angles 1.3. Some possible constructions This section presents two possible trisections. First we will see that any line segment can be trisected and then that there are some angles that can be trisected. Figure 1.3.1 1.3.1. Proposition. If OP is a given line segment, then using straightedge and compass alone it is possible to divide OP into three segments of equal length. Proof. Construct a second line segment OX, intersecting OP at O as in Figure 1.3.1. Using any radius r, draw a circle with center O and radius r let C be the point where this circle intersects OX. Now draw a circle of radius r with center C let B be the point where this circle intersects OX. Finally draw a third circle with radius r but centered at B let A be the point where this circle intersects OX. See Figure 1.3.1. So |OC| = |CB| = |BA| = r. Using the straightedge draw the segment AP . By Corollary 1.1.4 we can construct a line passing through B and parallel to AP let Q be the point where this parallel meets OP . Similarly construct a line parallel to AP and passing through the point C let R denote the point where this line meets OP . See Figure 1.3.2. Figure 1.3.2 By Proposition 1.2.1, ∠OAP = ∠OCR and ∠OPA = ∠ORC. Since the angle at O is shared by the triangles ΔOAP and ΔOCR, we have that these

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