Proof: By Euclid
(related to Proposition: 7.23: Divisor of One of Coprime Numbers is Coprime to Other)
 For if $C$ and $B$ are not prime to one another then [some] number will measure $C$ and $B$.
 Let it (so) measure (them), and let it be $D$.
 Since $D$ measures $C$, and $C$ measures $A$, $D$ thus also measures $A$.
 And ($D$) also measures $B$.
 Thus, $D$ measures $A$ and $B$, which are prime to one another.
 The very thing is impossible.
 Thus, some number does not measure the numbers $C$ and $B$.
 Thus, $C$ and $B$ are prime to one another.
 (Which is) the very thing it was required to show.
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"