Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Greater

Proof
Let $\rho$ be a rational straight line.

Let $m$ and $n$ be natural numbers such that $m^2 - n^2$ is not square.

Let $x$ be a straight line such that:
 * $(1): \quad m^2 : \paren {m^2 - n^2} = \rho^2 : x^2$

Thus:
 * $x^2 = \dfrac {m^2 - n^2} {m^2} \rho^2$

and so:
 * $x = \rho \sqrt {1 - k^2}$

where $k = \dfrac n m$.

From $(1)$:
 * $x^2 \frown \rho^2$

where $\frown$ denotes commensurability in length.

Thus $x$ is a rational straight line, but:
 * $x \smile \rho$

where $\smile$ denotes incommensurability in length.

From $(1)$:
 * $ m^2 : n^2 = \rho^2 : \rho^2 - x^2$

so that:
 * $\sqrt {\rho^2 - x^2} \frown \rho$

and in fact:
 * $\sqrt {\rho^2 - x^2} = k \rho$

where $k$ is a rational number.