How to proof that all points lie on a common circles.

Question

There are $n=2000$ points in the plane such that every for three of those points there exists a fourth of those points such that these four points lie on a common circle.

Proof that all $2000$ points lie on one common circle.

I can prove this for cases $n=5,6,7$ but I can't proof the case $n=2000$.

Thank you for helping.

Answer 1

An intuitive hint from me would be circumcenter and induction,

and you have to say something about collinear cases.