Plane Geometry - Bruce E. Shapiro

SECTION 45. EUCLIDEAN CONSTRUCTIONS 257Figure 45.7: Illustration of Construction 45.6, the perpendicular bisectorof a line segment.b, and c are constructable.Construction 45.11 (Rational Construction) Give a segment u = P Qof length 1, and any two integers p and q, construct a segment b that haslength p/q.1. Copy segment u a total of p times on ray −→ P Q.2. Define point A as the p th copy of point Q, so that P A = p.3. Construct any other ray −→ r emanating from P such that r is notparallel to or equal to −→ P A.4. Copy segment u a total of q times on ray −→ r .5. Define point U as the first copy of point Q on −→ r .6. Define point B as the p th copy of point Q on ray −→ r , so that P B = p.7. Construct a line l through Q 1 that is parallel to ←→ AB.8. Define point V = −→ P A ∩ l.9. UV = q/pConstruction 45.12 (Square Root) Given a segment u = P Q of length1, a second segment a = AB of length a, construct a segment that haslength √ a.1. Copy segment u onto line ←→ AB as segment u ′ = BC such that A∗B∗C.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.