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.

Answer 1

$$\frac{a}{b}=\frac{b}{c}$$

$$\frac{b}{a}=\frac{c}{b}$$

$$\frac{b}{a}+1=\frac{c}{b}+1$$

$$\frac{a+b}{a}=\frac{c+b}{b}=\frac{a}{b}$$

$$a^2-ba-b^2=0$$

$$a=\frac{b(1+\sqrt {5})}{2}$$

but

$$a=\frac{b(1-\sqrt {5})}{2}<0$$