Suppose otherwise. Let the coordinates of $A, B, C, A', B', C'$ be $(x_1, y_1), (x_2, y_2), (x_3,y_3), (a_1, b_1), (a_2, b_2),(a_3, b_3)$ respectively. For $i\neq j$, we have $(x_i-x_j)^2+(y_i-y_j)^2 \geq a^2$ and $(a_i-a_j)^2+(b_i-b_j)^2 \geq {a'}^2$, as well as $(x_i -a_j)^2+(y_i-b_j)^2 < \frac{a^2+{a'}^2}{3}$. Then $\sum_{i<j} (x_i-x_j)^2+(y_i-y_j)^2 + (a_i-a_j)^2+(b_i-b_j)^2 \geq 3(a^2+{a'}^2) >\sum_{i\neq j} (x_i -a_j)^2+(y_i-b_j)^2$. Rearranging, we get $0>((a_1+a_2+a_3)-(x_1+x_2+x_3))^2 +((b_1+b_2+b_3)-(y_1+y_2+y_3))^2$, which is clearly a contradiction.