Question 108
On a plane, the band with width $d$ is the set of all points whose distance from a line is less than or equal to $\frac{d}{2}$. There are four points $A$, $B$, $C$, $D$ on the plane. If you chose any three points among them, there exists a band with width $1$ containing them. Prove that there exists a band with width $\sqrt{2}$ containing all four points.