BALKAN-SL 2017 C4

For any set of points $A_{1}, A_{2}, \ldots, A_{n}$ on the plane, one defines $r\left(A_{1}, A_{2}, \ldots, A_{n}\right)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \geq 3$, there are indices $i, j, k$ such that

$$r\left(A_{1}, A_{2}, \ldots, A_{n}\right)=r\left(A_{1}, A_{,}, A_{k}\right)$$