Bumper bump

It's clear that $r(A_1, A_2, \cdots, A_n) \ge r(A_i, A_j, A_k)$ for any indices $i, j, k.$ In view of this, let $M = \max_{1 \le i < j < k \le n} r(A_i, A_j, A_k)$. We claim that $r(A_1, A_2, \cdots, A_n) = M.$ For each $1 \le i \le n$, let $\Omega_i$ denote the closed disc of radius $M$ centered at $A_i$. We know by $r(A_x, A_y, A_z) \le M, \forall 1 \le x < y < z \le n$ that any three of the $\Omega_i$'s have a common point. Hence, by Helly's theorem, they all share a common point $O$. The circle centered at $O$ of radius $M$ then contains all the points, and so $r(A_1, A_2, \cdots, A_n) \le M.$ It's clearly $\ge M$, so we're done. $\square$