# Product of Similar Plane Numbers is Square

## Theorem

In the words of Euclid:

If two similar plane numbers by multiplying one another make some number, the product will be square.

## Proof

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

$a : b = a^2 : a b$
there exists one mean proportional between $a$ and $b$.
there exists one mean proportional between $a^2$ and $a b$.
$a b$ is square.

$\blacksquare$