Equivalence of Definitions of Golden Mean

Theorem
The following definitions of the golden mean are equivalent:

Definition 1 is equivalent to Definition 3
Let $AB : AC = AC : BC$.

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

Then:

Definition 2 equivalent to Definition 3
Of these two roots, only $\dfrac {1 + \sqrt 5} 2$ is positive.