# Numbers between which exists one Mean Proportional are Similar Plane

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## Theorem

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 $20$)

## Proof

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 $b$.

Thus:

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

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 $p$

and

- $k q$ and $q$

respectively.

Then:

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

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

$\blacksquare$

## Historical Note

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

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{VIII}$. Propositions