Medial is Irrational

Proof
By definition, a medial is a mean proportional between two rational line segments which are commensurable in square only.

Let $\rho$ and $\rho \sqrt k$ be two rational line segments which are commensurable in square only.

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

From Commensurability of Elements of Proportional Magnitudes, $\rho^2 \sqrt k$ is incommensurable in length with $\rho^2$.

From, $\rho^2 \sqrt k$ is an irrational area.

Thus $\sqrt {\rho^2 \sqrt k} = \rho \sqrt[4] k$ is by definition an irrational straight line.

By definition of mean proportional, $\rho \sqrt[4] k$ is the medial between $\rho$ and $\rho \sqrt k$