# Golden Ratio in Wu Xing

### Solution

Counting the angles, we can reduce the problem with the straight lines to the simplified diagram:

Here, $\angle BOC=18^{\circ},\,$ $\angle COF=2\cdot 60^{\circ}+12^{\circ}=132^{\circ},\,$ so that $\angle GOF=60^{\circ}.$

Assuming the unit circle, $\displaystyle GO=\sin 18^{\circ}=\frac{\sqrt{5}-1}{4},\,$ $FO=\frac{\displaystyle GO}{\displaystyle \cos 60^{\circ}}=\displaystyle \frac{\sqrt{5}-1}{2}.\,$ It follows that

$\displaystyle \frac{FO}{EF}=\frac{FO}{1-FO}=\frac{\sqrt{5}-1}{3-\sqrt{5}}=\frac{\sqrt{5}+1}{2},$

the Golden Ratio.

For the intersections with the center circle, the problem reduces to the following diagram.

The required ratio $\displaystyle\frac{DG}{GO}=\varphi,\,$ is rather classical.

### Acknowledgment

The problem has been kindly posted on the CutTheKnotMath facebook page by Tran Quang Hung.

Disclaimer: Wu Xing is a five-element aspect of the Taoist thought, not necessarily represented by the above diagram.