jmc

We can write $$\frac{a+b}{c}+\frac{a+c}{b}+\frac{b+c}{a}=\frac{a}{c}+\frac{b}{c}+\frac{a}{b}+\frac{c}{b}+\frac{b}{a}+\frac{c}{a}.$$By AM-GM, $$\frac{a}{c}+\frac{b}{c}+\frac{a}{b}+\frac{c}{b}+\frac{b}{a}+\frac{c}{a}\ge 6\sqrt[6]{\frac{a}{c}\cdot \frac{b}{c}\cdot \frac{a}{b}\cdot \frac{c}{b}\cdot \frac{b}{a}\cdot \frac{c}{a}}=6.$$Equality occurs when $a=b=c,$ so the minimum value is $\boxed{6}.$