Construction of Sixth Binomial Straight Line

Proof

 * Euclid-X-53.png

Let $AC$ and $CB$ be straight lines constructed such that $AB = AC + CB$ is itself a straight line.

Let neither $AB : AC$ nor $AB : BC$ be the ratio which a square number has to a square number.

Let $D$ be a number which is not square.

Let $D$ have to neither $AB$ nor $AC$ the ratio that a square number has to another square number.

Let $E$ be a rational straight line.

Using, let:
 * $D : AB = E^2 : FG^2$

where $FG$ is a straight line.

From :
 * $E^2$ is commensurable with $FG^2$.

We have that $E$ is rational.

Therefore $FG$ is also rational.