Golden Ratio With Two Equal Circles And a Line

The following is an invention of Elliot McGucken.

Two equal circles centered at $B\;$ and $D\;$ and the radius $BC=DC.\;$ $F\;$ is the top point of $(B);\;$ line $FD\;$ crosses $(D)\;$ at $H\;$ and $I.$

Then $\displaystyle\frac{HI}{FH}=\varphi,\;$ the Golden Ratio.

The proof is as simple as the construction. Assume $BF=BC=BD=1.\;$ Then $FD=\sqrt{5},\;$ $HI=2,\;$ $DH=1,\;$ $FH=\sqrt{5}-1;\;$ $\displaystyle\frac{HI}{FH}=\frac{2}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{2}.$

Another simple construction in the same spirit has been found by Tran Quang Hung.

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