Question

N ≥ 3 points lie on a plane, but not all of them lie on a straight line. Prove that there is a circle passing through three of these points and containing none of the remaining points.

Answer 1

Let A and B be those of these points, the distance between which is the minimal. Then inside the circle with diameter AB there are no given points. Let C be that one of the remaining points from which the segment AB is seen under the largest angle. Then inside the circle passing through points A, B and C there are no given points.