Condition for Existence of Fourth Number Proportional to Three Numbers

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

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Historical Note

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

Sources

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