Proof: By Euclid
(related to Proposition: Prop. 9.31: Odd Number Coprime to Number is also Coprime to its Double)
 For if [$A$ and $C$] are not prime (to one another) then some number will measure them.
 Let it measure (them), and let it be $D$.
 And $A$ is odd.
 Thus, $D$ (is) also odd.
 And since $D$, which is odd, measures $C$, and $C$ is even, [$D$] will thus also measure half of $C$ [Prop. 9.30].
 And $B$ is half of $C$.
 Thus, $D$ measures $B$.
 And it also measures $A$.
 Thus, $D$ measures (both) $A$ and $B$, (despite) them being prime to one another.
 The very thing is impossible.
 Thus, $A$ is not unprime to $C$.
 Thus, $A$ and $C$ 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"