Between two Squares exists one Mean Proportional

Proof
Let $a^2$ and $b^2$ be square numbers.

Consider the number $a b$.

We have:
 * $\dfrac {a^2} {a b} = \dfrac a b = \dfrac {a b} {b^2}$

By definition, it follows that $a b$ is the mean proportional between $a^2$ and $b^2$.

Then:
 * $\paren {\dfrac a b}^2 = \dfrac {a^2} {b^2}$

By definition, it follows that $a^2$ has to $b^2$ the duplicate ratio that $a$ has to $b$.