Commensurability of Squares on Proportional Straight Lines

Proof
Let $A$, $B$, $C$ and $D$ be four straight lines in proportion, such that:
 * $A : B = C : D$

Using the lemma let the straight lines $E$ and $F$ be found such that:
 * $A^2 = B^2 + E^2$
 * $C^2 = D^2 + F^2$

As $A : B = C : D$ it follows from Similar Figures on Proportional Straight Lines that:
 * $A^2 : B^2 = C^2 : D^2$

But:
 * $E^2 + B^2 = A^2$
 * $F^2 + D^2 = C^2$

Therefore:
 * $E^2 + B^2 : B^2 = F^2 + D^2 : D^2$

and so by Magnitudes Proportional Compounded are Proportional Separated:
 * $E^2 : B^2 = F^2 : D^2$

By Similar Figures on Proportional Straight Lines:
 * $B : E = D : F$

But:
 * $A : B = C : D$

Therefore from Equality of Ratios Ex Aequali:
 * $A : E = C : F$

From Commensurability of Elements of Proportional Magnitudes it follows that:
 * if $A$ is commensurable with $E$ then $C$ is also commensurable with $F$

and:
 * if $A$ is incommensurable with $E$ then $C$ is also incommensurable with $F$.