Similar Plane Numbers have Same Ratio as between Two Squares

Proof
Let $a$ and $b$ be similar plane numbers.

From Between two Similar Plane Numbers exists one Mean Proportional, there exists a mean proportional $m$ between them.

By definition of mean proportional:
 * $\left({a, m, b}\right)$

is a geometric progression.

From Form of Geometric Progression of Integers:
 * $\exists k, p, q \in Z: a = k p^2, b = k q^2, m = k p q$

Thus:
 * $\dfrac a b = \dfrac {k p^2} {k q^2} = \dfrac {p^2} {q^2}$

Hence the result.