Construction of Components of Second Bimedial

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

Take the mean proportional $\rho \sqrt [4] k$ of $\rho$ and $\rho \sqrt k$, which is medial.

Let $x$ be such that:
 * $\rho \sqrt k : \rho \sqrt \lambda = \rho \sqrt [4] k : x$

which gives:
 * $x = \dfrac {\rho \sqrt \lambda} {\sqrt [4] k}$

We have that:
 * $\rho \sqrt k \frown \!\! - \rho \sqrt \lambda$

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

Thus:
 * $\rho \sqrt [4] k \frown \!\! - x$

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

Then:

Also see

 * Definition:Second Bimedial