A set of $n\gt3$ distinct points from plane $\pi$ satisfies the following criteria: for every subset of 3 points there is a circle of radius $R$ surrounding them. Prove that you can surround all points with a circle of radius $R$.
I have tried to encompass all points in a rectangle or a polygon, then tried to prove that such shape can be "covered" by a circle of the given radius, but the idea did not work. Also my attempt to make a proof by induction fell short on the induction step.
Consider a circle with minimal radius $r$ that contains all the points. At least three points lie on this circle (else we could shrink it), and at least three points that lie on this circle are not on the same semicircle (else we could shrink it). These three points are not contained in any circle of radius less than $r$, but by the premise they're contained in a circle of radius $R$. Hence $r\le R$.