Magnitudes with Rational Ratio are Commensurable/Porism

Porism to Magnitudes with Rational Ratio are Commensurable

In the words of Euclid:

From this it is manifest that, if there be two numbers, as $D$, $E$, and a straight line, as $A$, it is possible to make a straight line [$F$] such that the given straight line is to it as the number $D$ to the number $E$.
And, if a mean proportional be also taken between $A$, $F$, as $B$,
as $A$ is to $F$, so will the square on $A$ be to the square on $B$, that is, as the first is to the third, so is the figure on the first to that which is similar and similarly described on the second.
But, as $A$ is to $F$, so is the number $D$ to the number $E$; therefore it has been contrived that, as the number $D$ is to the number $E$, so also is the figure on the straight line $A$ to the figure on the straight line $B$.

(The Elements: Book $\text{X}$: Proposition $6$ : Porism)

Proof

Apparent from the construction.

$\blacksquare$

Historical Note

This proof is Proposition $6$ of Book $\text{X}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions