prove the ratio of the line in the question

Question

the book's solution uses $(a+b)/a= a/b$. I got my answer using b and c but want to understand the book's method.

Lion Heart · Accepted Answer

$$frac{a}{b}=frac{b}{c}$$
$$frac{b}{a}=frac{c}{b}$$
$$frac{b}{a}+1=frac{c}{b}+1$$
$$frac{a+b}{a}=frac{c+b}{b}=frac{a}{b}$$
$$a^2-ba-b^2=0$$
$$a=frac{b(1+sqrt {5})}{2}$$
but
$$a=frac{b(1-sqrt {5})}{2}<0$$

prove the ratio of the line in the question

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