Plane Geometry - Bruce E. Shapiro

256 SECTION 45. EUCLIDEAN CONSTRUCTIONSFigure 45.6: Construction 45.5: dropping of a perpendicular to a line froma point not on the line. The bisector of ∠AP B is perpendicular to line l.segment n times we can construct a segment of any integer length n.Definition 45.10 A number x is said to be constructible if we can constructa segment of length x given a segment of length 1, with a straightedgeand compass.Are all numbers constructible? If not, which numbers are?1. Any integer length n can be constructed by duplicating the unit segmentn times.2. Given any segments of length p and q, where p > q, both integers, wecan find segments of length p + q, p − q, p ∗ q (construction ??), andany combination thereof.3. By finding midpoints repeatedly, we can find a segment that is 1/2 mtimes as long as any of these other segments.4. By adding these segments together, we can find segments of lengthk/2 m for k = 1, 2, ..., 2 m .5. We can find segments of any rationl length p/q, where p, q are anyintegers (construction 45.11).6. For any segment of length x, we can find a segment of length √ x(construction 45.12).7. We can construct elemnts of the extension field having the field ofrational numbers as a subset.8. We can construct elements of the extension field of a + b √ c, where a,« CC BY-NC-ND 3.0. Revised: 18 Nov 2012