Condition for Existence of Fourth Number Proportional to Three Numbers

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Theorem

In the words of Euclid:

Given three numbers, to investigate whether it is possible to find a fourth proportional to them.

(The Elements: Book $\text{IX}$: Proposition $19$)


Proof


Historical Note

This theorem is Proposition $19$ of Book $\text{IX}$ of Euclid's The Elements.


Sources