jmc

Let $r$, $s$, and $t$ be the radii of the circles centered at $A$, $B$, and $C$, respectively. Then $r+s=3$, $r+t=4$, and $s+t=5$. Adding these three equations gives $2(r+s+t)=12$, so $r+s+t=6$. Combining this with the original three equations, we get $r=1$, $s=2$, and $t=3$. Thus the sum of the areas of the circles is $$\pi (1^2+2^2+3^2)=\boxed{14\pi }.$$