Golden Ratio in Three Regular Pentagons

Problem

Golden Ratio in Three Regular Pentagons, problem

Solution

Solutions are welcome.

Extra

The configuration of three regular pentagons proved beckoning and promising. After a misstep that used only two pentagons, $\displaystyle \frac{FG}{AF}=\varphi,\,$ another occurrence of the Golden Ratio, authentically in three pentagons has popped up:

Let $AD\,$ intersect $LK\,$ in $O.\,$

Golden Ratio in Three Regular Pentagons, extra occurrence

Then $\displaystyle \frac{AD}{DO}=\varphi,\,$ the Golden Ratio.

Acknowledgment

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

 

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