The title says it all.
If $\gcd(A, B) = 1$, what sufficient condition(s) (if any) guarantee $\gcd(A, C) = \gcd(B, C) = 1$?
For example, $A = 3$, $B = 4$ and $C = 5$ satisfy the constraints.
Are there any other examples?
(Please feel free to remove the pythagorean-triples tag if it is not appropriate for this question.)