Equivalence of Definitions of Golden Mean

Theorem

The following definitions of the concept of Golden Mean are equivalent:

Definition 1

Let a line segment $AB$ be divided at $C$ such that:

$AB : AC = AC : BC$

Then the golden mean $\phi$ is defined as:

$\phi := \dfrac {AB} {AC}$

Definition 2

The golden mean is the unique positive real number $\phi$ satisfying:

$\phi = \dfrac {1 + \sqrt 5} 2$

Definition 3

The golden mean is the unique positive real number $\phi$ satisfying:

$\phi = \dfrac 1 {\phi - 1}$

Proof

Definition 1 is equivalent to Definition 3

Let $AB : AC = AC : BC$.

Let $\dfrac {AB} {AC} = \dfrac {AC} {BC} = \phi$.

Then:

 $\ds \phi$ $=$ $\ds \frac {AC + BC} {AC}$ as $AB = AC + BC$ $\ds$ $=$ $\ds 1 + \frac {BC} {AC}$ $\ds$ $=$ $\ds 1 + \frac 1 \phi$ $\ds \leadstoandfrom \ \$ $\ds \phi - 1$ $=$ $\ds \frac 1 \phi$ $\ds \leadstoandfrom \ \$ $\ds \frac 1 {\phi - 1}$ $=$ $\ds \phi$

$\Box$

Definition 2 equivalent to Definition 3

 $\ds \phi$ $=$ $\ds \frac 1 {\phi - 1}$ Definition 3 of Golden Mean $\ds \leadstoandfrom \ \$ $\ds \phi \paren {\phi - 1}$ $=$ $\ds 1$ $\ds \leadstoandfrom \ \$ $\ds \phi^2 - \phi - 1$ $=$ $\ds 0$ $\ds \leadstoandfrom \ \$ $\ds \phi$ $=$ $\ds \frac {1 \pm \sqrt {1^2 - 4 \times 1 \times \paren {-1} } } 2$ Quadratic Formula $\ds$ $=$ $\ds \frac {1 \pm \sqrt 5} 2$

Of these two roots, only $\dfrac {1 + \sqrt 5} 2$ is positive.

$\blacksquare$