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 CX / CD = φ.

Assume for simplicity that AB = 2. Then CD = 23, CX = 15 + 3, so that

CX / CD= (15 + 3) / 23
 = (5 + 1) / 2
 = φ.

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.

References

  1. K. Hofstetter, A Simple Construction of the Golden Section, Forum Geometricorum, v 2 (2002), pp. 65-66
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