Golden Ratio by Compass Only
In a 2002 article, K. Hofstetter, offered an elegant way of obtaining the Golden Ratio.
It will be convenient to denote S(R) the circle with center S through point R. For the construction, let A and B be two points. Circles A(B) and B(A) intersect in C and D and cross the line AB in points E and F. Circles B(E) and A(F) intersect in X and Y, as in the diagram. Because of the symmetry, points X, D, C, Y are collinear. The fact is
Assume for simplicity that AB = 2. Then
|CX / CD||= (√15 + √3) / 2√3|
|= (√5 + 1) / 2|
Also, CD / DX = φ. Finally, observe that points E and F lie on C(D). It follows that the whole construction can be accomplished with compass only.
- K. Hofstetter, A Simple Construction of the Golden Section, Forum Geometricorum, v 2 (2002), pp. 65-66