Question 47
For any set of points $A_1, A_2, ..., A_n$ on the plane, one defines $r(A_1, A_2, ..., A_n)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \ge 3$, there are indices $i, j, k$ such that
$$
r(A_1, A_2, ..., A_n) = r(A_i, A_j, A_k).
$$
$$
r(A_1, A_2, ..., A_n) = r(A_i, A_j, A_k).
$$