Proposition: 8.03: Geometric Progression in Lowest Terms has Coprime Extremes
(Proposition 3 from Book 8 of Euclid's “Elements”)
If there are any multitude whatsoever of numbers in continued proportion (which are) the least of those (numbers) having the same ratio as them then the outermost of them are prime to one another.
Modern Formulation
If $\frac AB=\frac BC=\frac CD$ are a geometric progression and the least of these numbers, then $A$ and $D$ are coprime.
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016