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Old 2020-01-14, 14:45   #3
Dr Sardonicus
 
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The usual formulation for x and y being in golden proportion is \frac{x}{y}\;=\;\frac{x\;+\;y}{x}; the right-hand side clearly is greater than 1. Taking y = 1 gives

\frac{x}{1}\;=\;\frac{x\;+\;1}{x}\text{, or }x^{2}\;-\;x\;-\;1\;=\;0\text{.}

An illustration is given by the 72-72-36 degree isosceles triangle. The bisector of one of the 72-degree angles divides the opposite side in golden ratio; calling x the length of the base and y the length of the smaller segment of the side opposite the angle bisector, gives the above proportion.
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