# Does there always exist a circle through three points such that any other points inside (or lie on) the circle?

Could You help me give a proof that:

Given a finite number $(>3)$ of points in the Euclidean plane, then exist a circle through three points such that any other points inside (or lie on) the circle.

• You would obviously need to assume that the points do not all lie on a straight line – Walt van Amstel Jun 8 '17 at 14:22
• Two ideas that I don't know how to make rigorous. (1)Start with a circle that contains all the points and keep shrinking it down and show that it will stop only when there's three points in the set. (2) Consider all circles through each set of three points in the convex Hull and show that one of them will contain all the points. – marty cohen Jun 8 '17 at 14:35
• Thank to You, maybe is this an aswer? – Cố Gắng Lên Jun 8 '17 at 14:40
• have a look to (en.wikipedia.org/wiki/Smallest-circle_problem). – Jean Marie Jun 8 '17 at 15:47
• @Jean Marie: The minimum bounding circle doesn't necessarily pass through 3 of the points. – Jens Jun 8 '17 at 15:52