Construction of Components of First Bimedial

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.

Also see

 * Definition:First Bimedial