Euclidean Algorithm/Euclid's Proof/Porism

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Porism to Euclidean Algorithm

In the words of Euclid:

From this it is manifest that, if a number measure two numbers, it will also measure their greatest common divisor.

(The Elements: Book $\text{VII}$: Proposition $2$ : Porism)


Proof

An algebraic proof of this is given in Common Divisor Divides GCD.

$\blacksquare$


Historical Note

This theorem is Proposition $2$ of Book $\text{VII}$ of Euclid's The Elements.


Sources