Between two Similar Plane Numbers exists one Mean Proportional

Proof
Let $m$ and $n$ be similar plane numbers.

Then for some $p_1, p_2, q_1, q_2 \in \Z$ such that $p_1 < p_2$ and $q_1 < q_2$:


 * $m = p_1 p_2$
 * $n = q_1 q_2$

such that:
 * $\dfrac {p_1}{q_1} = \dfrac {p_2}{q_2}$

Thus let:
 * $r := p_1 q_2 = q_1 p_2$

So:

and:

Thus by definition, $\paren {m, r, n}$ is a geometric sequence.

By definition, $r$ is a mean proportional between $m$ and $n$.

Thus as:
 * $\dfrac {p_1}{q_1} = \dfrac m r = \dfrac r n = \dfrac {p_2}{q_2}$

it follows that $m$ is in duplicate ratio to $n$ as their sides.