Golden Ratio in Pentagon And Three Triangles

Tran Quang Hung
December 31, 2016

Golden Ratio in Pentagon And Three Triangles, problem

 

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

Golden Ratio in Pentagon And Three Triangles, proof

We have, $FN=ON\sin 72^{\circ},\,$ $OF=ON\cos 72^{\circ},\,$ $FT=OF\tan 36^{\circ}.\$ We are interested in the ratio

$\displaystyle\begin{align} \frac{SN}{TN}&=\frac{FN+SF}{FN-FT}=\frac{FN+FT}{FN-FT}\\ &=\frac{\sin 72^{\circ}+\cos 72^{\circ}\tan 36^{\circ}}{\sin 72^{\circ}-\cos 72^{\circ}\tan 36^{\circ}}=\frac{\sin 72^{\circ}\cos 36^{\circ}+\cos 72^{\circ}\sin 36^{\circ}}{\sin 72^{\circ}\cos 36^{\circ}-\cos 72^{\circ}\sin 36^{\circ}}\\ &=\frac{\sin 108^{\circ}}{\sin 36^{\circ}}=\frac{\sin 72^{\circ}}{\sin 36^{\circ}}=\frac{2\sin 36^{\circ}\cos 36^{\circ}}{\sin 36^{\circ}}\\ &=2\cos 36^{\circ}=\frac{\sqrt{5}+1}{2}\\ &=\varphi, \end{align}$

because $\displaystyle\cos 36^{\circ}=\frac{\sqrt{5}+1}{4},\,$ and using the sine of the sum formula.

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