suppose d|a and d|(a+b). then surely d divides b = (a+b) - a.
on the other hand, suppose k|b and k|(a+b). then surely k divides (a+b) - b = a.
thus shows that the sets of common divisors of a and a+b and b and a+b are both contained in the set of common divisors of a and b.
but if t|a and t|b, then t divides a+b as well, showing that the set of common divisors of a and b is contained in both sets of common divisors of (a,a+b) and (b,a+b).
hence all 3 sets are equal.
an example: let a = 60 and b = 84. then gcd(60,84) = 12. and, as expected, gcd(60,144) = 12, and gcd(84,144) = 12.