Construction of Components of First Bimedial

Theorem

In the words of Euclid:

To find medial straight lines commensurable in square only which contain a rational rectangle.

Proof

Let $\rho$ and $\rho \sqrt k$ be rational straight lines which are commensurable in square only.

Their mean proportional is $\rho \sqrt [4] k$ which is medial.

Let $x$ be such that:

$\rho : \rho \sqrt k = \rho \sqrt [4] k : x$

which gives:

$x = \rho k^{3/4}$

We have that:

$\rho \frown \!\! - \rho \sqrt k$

where $\frown \!\! -$ denotes commensurability in square only.

Thus:

$\rho \sqrt [4] k \frown \!\! - \rho k^{3/4}$

From Straight Line Commensurable with Medial Straight Line is Medial it follows that $\rho k^{3/4}$ is also medial.

$\blacksquare$

Historical Note

This proof is Proposition $27$ of Book $\text{X}$ of Euclid's The Elements.