Show that if $n$ points are such that any three lie in a circle of radius $1$, then all of them lie in a circle of radius $1$

Question

Consider a set of $n$ points in the plane such that any three of them are contained in a circle with radius $r=1$. Prove by induction that all $n$ points are contained in a circle with radius $r=1$.

Answer 1

Consider an "envelope" of the points, which form a convex polygon. We choose the maximum angle of a convex polygon. We describe a circle around the three neighboring вершина that form the angle. All the other points will lie within this circle.

Answer 2

If any three of the $n$ points are contained in a circle with radius $r=1$, then the distance between any two of the $n$ points is at most $d=2$. Hence the diameter of the $n$ points as a set is at most $2$. In particular they are contained in a circle of diameter $2$, which has radius $r=1$.