Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square

Proof
Let $A$ and $B$ be two medial straight lines.

Let $A$ and $B$ be commensurable in square only.

Then their lengths are of the form:
 * $l \left({A}\right) = \rho \sqrt [4] k$
 * $l \left({B}\right) = \rho \sqrt [4] k \sqrt \lambda$

where:


 * $\rho$ is a rational number
 * $k$ and $\lambda$ are rational numbers whose square root is irrational.

The rectangle contained by $A$ and $B$ has area $R$ given by:


 * $R = \rho^2 \sqrt k \sqrt \lambda$

In general, $R$ is medial.

However, let $\sqrt \lambda = k' \sqrt k$ where $k$ is rational.

Then:
 * $R = \rho^2 \sqrt k k' \sqrt k = \rho^2 k k'$

which is rational.