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Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Coprime to other Element
Euclid's Formulation
If three numbers in continued proportion are the least of those (numbers) having the same ratio as them then two (of them) added together in any way are prime to the remaining (one).
 Let $A$, $B$, $C$ be three numbers in continued proportion (which are) the least of those (numbers) having the same ratio as them.
 I say that two of $A$, $B$, $C$ added together in any way are prime to the remaining (one), (that is) $A$ and $B$ (prime) to $C$, $B$ and $C$ to $A$, and, further, $A$ and $C$ to $B$.
Modern Formulation
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Table of Contents
Proofs: 1
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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