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I want to prove: if $z$ is the right golden ratio point and $u$ the left golden ratio point to an interval $[x,y]$, then $z$ is the left golden ratio point to $[u,y]$.

Definition - Golden ratio points $\alpha_1,\alpha_2$ for an interval $[x,y]$ are solutions to $$\frac{y-x}{\alpha_1-x}=\frac{\alpha_1-x}{y-\alpha_1},$$ $$\frac{y-x}{\alpha_2-x}=\frac{\alpha_2-x}{y-\alpha_2}.$$

An ideas?

  • 0
    could you state the correct definition? – 2017-01-19
  • 0
    It should be correct now. Sorry about the typo. – 2017-01-19
  • 0
    Maybe add the subscripts for $\alpha$. – 2017-01-19

1 Answers 1

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Points $U,Z$ lie symetrically, so let us denote $a = |XU| = |ZY|$, $b = |UZ|$. By assumptions, $$ \frac{a}{a+b} = \frac{a+b}{2a+b}. $$ What we want is that $$ \frac{b}{a} = \frac{a}{a+b}. $$ These two are easily seen to be equivalent.