a+b/c+d=√a2+b2/√c2+d2

1.
令a/b=c/d=t,则a=bt c=dt
(a+b)/(c+d)=(bt+b)/(dt+d)=b(t+1)/d(t+1)=b/d
√(a²+b²)/√(c²+d²)=√(b²t²+b²)/√(d²t²+d²)=√[b²(1+t²)]/√[d²(1+t²)]=√b²/√d²
b>0 d>0 √(a²+b²)/√(c²+d²)=b/d
(a+b)/(c+d)=√(a²+b²)/√(c²+d²)
因此a/b=c/d且b、d均为正数是等式成立的充分条件.
2.
令a/b=c/d=t,则a=bt c=dt
(a+b)/(c+d)=(bt+b)/(dt+d)=b(t+1)/d(t+1)=b/d
√(a²+b²)/√(c²+d²)=√(b²t²+b²)/√(d²t²+d²)=√[b²(1+t²)]/√[d²(1+t²)]=√b²/√d²