Euclidean Construction of the Golden Ratio The Pythagorean relation leads to a simple cons...

Euclidean Construction of the Golden Ratio The Pythagorean relation leads to a simple construction of the Golden Ratio τ, a number that may be defined in terms of the radical expression ≈ 1.618

(a) Let a be the length of a given segment . Construct the midpoint M of segment , and a square ABCD on as side. With M as center and as radius, swing an arc, cutting line at point E. Show that the ratio AE/AD is the Golden Ratio.

*See Problem 15, Section 4.3 and Problems 16, 17, Section 4.4.

(b) Show that the Golden Ratio τ satisfies the defining relation τ2 = τ + 1. (Show that τ is the positive root of the quadratic equation .x2 - x - 1 = 0.)

(c) Show that τ -2 = 2 - τ.