Numbers between which exists one Mean Proportional are Similar Plane

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Let $a, b \in \Z$ such that the geometric mean is an integer.

Then $a$ and $b$ are similar plane numbers.

In the words of Euclid:

If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.

(The Elements: Book $\text{VIII}$: Proposition $20$)


Let the geometric mean of $a$ and $b$ be an integer $m$.

Then, in the language of Euclid, $m$ is a mean proportional of $a$ and $b$.


$\left({a, m, b}\right)$

is a geometric sequence.

From Form of Geometric Sequence of Integers:

$\exists k, p, q \in \Z: a = k p^2, b = k q^2$

So $a$ and $b$ are plane numbers whose sides are:

$k p$ and $p$


$k q$ and $q$



$\dfrac {k p} {k q} = \dfrac p q$

demonstrating that $a$ and $b$ are similar plane numbers by definition.


Historical Note

This proof is Proposition $20$ of Book $\text{VIII}$ of Euclid's The Elements.