Golden Ratio in Pentagon And Two Squares

Tran Quang Hung

December 24, 2016

In the dagram below, $\displaystyle\frac{MS}{ST}=\varphi,\,$ the Golden Ratio.

For a proof, add a couple of lines to the diagram:

By the Law of Sines in $\Delta MOS,\,$

$\displaystyle\frac{MS}{OS}=\frac{\sin \angle MOS}{\sin\angle OMS}=\frac{\sin 54^{\circ}}{\sin 18^{\circ}}.$

In $\Delta JOS,\,$

$\displaystyle\frac{OS}{JS}=\frac{1}{\sin 54^{\circ}}.$

Taking the product,

$\displaystyle\frac{MS}{ST}=\frac{MS}{2\cdot JS}=\frac{1}{2\sin 18^{\circ}}=\frac{2}{\sqrt{5}-1}=\frac{\sqrt{5}+1}{2}=\varphi$

because $\displaystyle\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}.$

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