There exist two points $P$ and $Q$ in $A$

Question

I would appreciate if somebody could help me with the following problem

Q: Let $A$ be a set of $n$ distinct points in $\mathbb{R}^2$. Prove that there exist two points $P$ and $Q$ in $A$

such that

(1). $P\neq Q$

(2). At least there are $\lfloor \frac{n}{3} \rfloor$ distinct points on and inside the circle $S$ with center $\frac{P+Q}{2}$and radius $\frac{|P-Q|}{2}$.

Answer 1

Take P1 and Q1 be the most distant points in A and draw the circle than has P1,Q1 for diameter. (grey circle)

All the points in A lie in the circle centered on P1 and with radius P1-Q1 and in a third circle centered on Q1 and with the same radius, because of the maximum distance choice. (circles yellow and green)

Then A is at the intersection of the last 2 circles, which contains the first one. This defines 3 zones. At least one of them has more than n/3 points.

If it is the initial P1/Q1 circle you win, else ... I don't know yet but maybe it is a good start :)