Is gcd(a, b+c) equal to gcd(gcd(a, b), gcd(a, c))

Question

Is there any relation between gcd(a, b+c) and gcd(gcd(a, b), gcd(a, c)). I have proved that gcd(gcd(a, b), gcd(a, c)) divides gcd(a, b+c). But, what is exact gcd(a,b+c)

Answer 1

Using the shorthand notation $(x,y)$ for gcd$(x,y)$ one result connecting $(a,b),(a,c)$ and $(a,b+c)$ is the simple $$((a,b)(a,c))=((a,c)(a,b+c))=((a,b+c)(a,b)).$$