Plane Geometry - Bruce E. Shapiro

258 SECTION 45. EUCLIDEAN CONSTRUCTIONSFigure 45.8: Construction 45.7, construction of a parallel line through apoint that is not on the line.2. Find the midpoint M of AC.3. Construct circle γ = C(M, AM).4. Construct a line l through B that is perpendicular to AC.5. γ intersects l at two points. Name one of these points D. (It doesnot matter which one).6. Let c = BD. Then c = √ a.Theorem 45.13 All constructible numbers are algebraic numbers, i.e., everyconstructible number is a root of the algebraic equationwhere a 0 , a 1 , ..., a n are integers.a n x n + a n−1 x n−1 + ... + a 0 = 0A number is not algebraic is said to be transcendental. Since π is transcendental1 , we have the following result.Theorem 45.14 π is not constructible.Corollary 45.15 The construction of squaring the circle with a straightedgeand a compass is impossible.Theorem 45.16 The general construction of trisecting an angle cannot beperformed with a straight-edge an compass alone.1 This result was proven by T. Lindemann in 1882.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012