Golden Ratio With Two Unequal Circles And a Line

The following invention of Tran Quang Hung, was likely inspired by an earlier construction of Elliot McGucken.

Given circle $(O)\;$ diameter $AB.\;$ Construct the circle $(K)\;$ diameter $OB.\;$ Let $C\;$ be the midpoint of arc $AB;\;$ $M,N\;$ the intersections of $CK\;$ with $(K).\;$

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Golden Ratio With Two Unequal Circles And a Line

Then $M\;$ divides $CN\;$ in the Golden Ratio: $\displaystyle\frac{MN}{CM}=\varphi.$

The proof is as simple as the construction. Assume $OK=KM=KN=1.\;$ Then $CK=\sqrt{5},\;$ $MN=2,\;$ $CM=\sqrt{5}-1;\;$ $\displaystyle\frac{MN}{CM}=\frac{2}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{2}.$

Xuan Thang Vu offered a different approach: $CO,\;$ being a tangent to $(K),\;$ $CO^2=CM\cdot CN,\;$ or, say, $a^2=x(x+a).\;$ With $\displaystyle y=\frac{a}{x},\;$ $y^2=y+1,\;$ making $y=\varphi.$

Tran Quang Hung has observed additional occurrencies of the Golden Ratio that could be surmised in the derived diagram:

Golden Ratio With Two Unequal Circles And a Line, Extra

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