First year french college. Is there a finite set of points of the plane A. Finite set of point A such as... Is there a finite set of at least 3 points in the plane A, such as every 3 points of A are not aligned, and such as every circle going through 3 points of more of A, it’s center is in A ?
I believe that there can’t be such a set. So I tried showing that for every n point of the plane you can’t arrange them to be set A.
Being based on a number n I also tried doing it by doing a « mathematical induction » to show that there can’t be a set A for n points.
I’m a french student in my first year of college, I’m sorry by advance if my english is a bit wacky.
edit: forgot that it's a set of at least 3 points sorry
Answer to be proved in details
Such a finite set of points $A$ can't exist.
Denote by $P_1, P_2, P_3$ three points for which the circumscribed circle $\mathcal C$ is achieving the minimum radius $R$ of all circumscribed circles. Let $O$ be its center. By hypothesis $O \in A$.
Assumption to be proven: one of the triangles $(O \ P_1 \ P_2)$, $(O \ P_2 \ P_3)$ or $(O \ P_1 \ P_3)$ has a circumscribed radius strictly less than $R$. A contradiction.
In the plane draw a circle, take  three points on the circle (I think you can take any  finite number of points but 3 will do). Call these points your set A. These 3 points are not aligned (not on the same line) and there is only one circle that goes through them and we know by construction that this circle is in the plane.