I have solved this following Combinatorial Geometry Problem using extremal principle.Please check whether this solution is correct or not.Also write if you have any other solution.
Problem :-
Let $S$ be a finite set of points $(|S|≥3)$ in the Euclidean plane such that
$(1)$ No three points are collinear
$(2)$ If $A,B,C$ are in $S$ then $A',B',C'$ are also in $S$ where $A',B',C'$ are diametrically opposite points of $A,B,C$ with respect to the circumcircle of $∆ABC$. Then prove that all the points in $S$ are concyclic.
Solution :-
Let's call a circumcircle of some triangle formed by some three points in $S$ Maximal if it contains all other points in $S$ either inside it or on the perimeter of it.
$Lemma:$ There exist three points in $S$ such that the corresponding circumcircle of the triangle determined by these three points is $Maximal$.
$Proof:$ Let $|S|=n$ and let us write $S$ as $S=\{P_1,P_2,P_3,...,P_n\}$
Take any two points in $S$ such that all other points lie on the same side of the line determined by these two points.W.l.o.g. we can take these two points to be $P_1$ and $P_2$.Now consider the angles $〈P_1P_kP_2$ with $k=3,4,...,n$.Now among this finite number of angles there exists a minimum.Again w.l.o.g. let this minimum angle be $〈P_1P_3P_2$.Then draw the circumcircle of triangle $P_1P_2P_3$.Clearly all the points of $S$ lie within this circle.So this is the desired $Maximal$ circle.Hence we have proved the existence of a $Maximal$ circle.Let us name this circle $C$.
Now we have to prove that all the points of $S$ lie on the perimeter of $C$.According to the property of $C$ no point of $S$ lies outside $C$.Then we have to prove that no point of $S$ lies inside $C$.Suppose to the contrary that there exists at least one point $D$ of $S$ which lies inside $C$.Now $P_1$ lies on the perimeter of $C$ implies $P_1'$ lies on the perimeter of $C$ where $P_1P_1'$ is a diameter of $C$.Obviously the circumcircle of the triangle $P_1DP_1'$ is bigger than $C$ and the point $D$ and its center lies on different sides of $P_1P_1'$.Now according to condition $(2)$ given in question the diametrically opposite point of $D$ say $D'$ also lies in $S$.But $D'$ lies on the complementary arc of the arc $P_1DP_1'$ which lies completely outside $C$.Hence $D'$ lies outside $C$.Hence we found a point in $S$ which lies outside $C$.But this is contradictory to the fact that $C$ is $Maximal$.Hence all points of $S$ lies on the perimeter of $C$.So all points of $S$ are concyclic.