## Intersection of balls with same radius but moving the centers

Let $B_i(a_i,r_i)$ a set of m balls in the euclidean space $R^n$, with
center $a_i$ and radius $r_i$. Let us suppose that they have non-empty
intersection.

Now consider new centers $b_i$ such that $|b_i-b_j|<|a_i-a_j|$. Then
¿how can I prove that the balls $B_i(b_i,r_i)$ have also non-empty
intersection?

I have tried by taking the same baricentric coordiantes respect to the centers $a_i$ and later with centers $b_i$. But this doesn't work.

Thank you very much for your help.