Proof: By Euclid
(related to Proposition: 7.28: Numbers are Coprime iff Sum is Coprime to Both)
 "$\Rightarrow$"
 Let the two numbers, $AB$ and $BC$, (which are) prime to one another, be laid out.
 Assume, $CA$ and $AB$ are not prime to one another.
 Then some number will measure $CA$ and $AB$.
 Let it (so) measure (them), and let it be $D$.
 Therefore, since $D$ measures $CA$ and $AB$, it will thus also measure the remainder $BC$.
 And it also measures $BA$.
 Thus, $D$ measures $AB$ and $BC$, which are prime to one another.
 The very thing is impossible.
 Thus, some number cannot measure (both) the numbers $CA$ and $AB$.
 Thus, $CA$ and $AB$ are prime to one another.
 So, for the same (reasons), $AC$ and $CB$ are also prime to one another.
 Thus, $CA$ is prime to each of $AB$ and $BC$.
 "$\Leftarrow$"
 So, again, let $CA$ and $AB$ be prime to one another.
 For if $AB$ and $BC$ are not prime to one another then some number will measure $AB$ and $BC$.
 Let it (so) measure (them), and let it be $D$.
 And since $D$ measures each of $AB$ and $BC$, it will thus also measure the whole of $CA$.
 And it also measures $AB$.
 Thus, $D$ measures $CA$ and $AB$, which are prime to one another.
 The very thing is impossible.
 Thus, some number cannot measure (both) the numbers $AB$ and $BC$.
 Thus, $AB$ and $BC$ are prime to one another.
 (Which is) the very thing it was required to show.
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"