Greatest Common Measure of Three Commensurable Magnitudes/Porism

Porism to Greatest Common Measure of Three Commensurable Magnitudes

In the words of Euclid:

From this it is manifest that, if a magnitude measure three magnitudes, it will also measure their greatest common measure.
Similarly too, with more magnitudes, the greatest common measure can be found, and the porism can be extended.

(The Elements: Book $\text{X}$: Proposition $4$ : Porism)

Proof

Apparent from the construction.

$\blacksquare$

Historical Note

This proof is Proposition $4$ 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