1.5. Constructible points and constructible numbers 17 and radius 1 2 (1 + a). So R = ( 1 2 (1 + a), 0) is the point where it meets the positive x-axis. Construct the line perpendicular to the x-axis and passing through the point ( 1 2 (1 a), 0). Let P and Q be the points where the circle meets above and below the x-axis, respectively. See Figure 1.5.3. Figure 1.5.3 If P = ( 1 2 (1 a),y), then, by virtue of being on the circle, 1 2 (1 a) 2 + y2 = 1 4 (1 + a)2 Solving for y we see that the second coordinate of P is a. The reader might stand back and contemplate what we have accom- plished in this section. We have that every rational number is constructible and every square root of a constructible number is constructible. Therefore numbers such as 2 + 11 17 1 19 + 5 8 1 7 + 2 5 are constructible. In other words, every number that can be expressed using sums, differences, products, quotients, and square roots of a rational number is constructible. Eventually we will show that every constructible number can be so expressed. How to say this precisely and how to prove it requires some more algebra, which we will develop shortly. We can now precisely state the Trisection Problem. Say that an angle is constructible if the rays forming the angle are parts of constructible lines in other words, it is formed by two constructible lines.
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