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Proof: By Euclid

(related to Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square)

• Let $A$, $B$, $C$ be three numbers in continued proportion, and let the first $A$ be square.

• I say that the third $C$ is also square.

• For since one number, $B$, is in mean proportion to $A$ and $C$, $A$ and $C$ are thus similar plane (numbers) [Prop. 8.20].

• And $A$ is square.
• Thus, $C$ is also square [Def. 7.21] .
• (Which is) the very thing it was required to show.
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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"