TITLE: Golden ratio problem
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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.