Proof: By Euclid
(related to Proposition: 7.25: Square of Coprime Number is Coprime)
 For let $D$ be made equal to $A$.
 Since $A$ and $B$ are prime to one another, and $A$ (is) equal to $D$, $D$ and $B$ are thus also prime to one another.
 Thus, $D$ and $A$ are each prime to $B$.
 Thus, the (number) created from (multilying) $D$ and $A$ will also be prime to $B$ [Prop. 7.24].
 And $C$ is the number created from (multiplying) $D$ and $A$.
 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"