$ab-(a+b)(a-b)=0$ and the Golden ratio.

Question

I have found:

$b=a*\phi$

$b=a*(-\phi)$

$b=a/\phi$

Trying to find the correlation with the equation and phi, any insight how to demonstrate this or a proof?

Answer 1

The golden ratio $\phi$ is usually defined to be the ratio of $a$ and $b$ as in this diagram:

where the red rectangle, with the proportions $\frac ab$, is similar to the large rectangle with proportions $\frac{a+b}a$, so $$\phi = \frac ab = \frac{a+b}{a}.$$

Multiplying on both sides by $ab$ we obtain $$a^2 = ab+b^2$$ so $$ab = a^2 - b^2 = (a+b)(a-b).$$