More generally, we may prove the following result : Let $S$ be a set of $2k+1$ points in the plane, no three collinear, no four concyclic. A circle is said to be an halving circle if it has three points of $S$ on its circumference and exactly $k-1$ points of $S$ in its interior (and thus exactly $k-1$ in its exterior). Thus, the number of halving circles is $k^2$. Reference : F. Ardila, The number of halving circles, A.M.M. 2004, p.586-591. Pierre.

Please give a solution here, either for the extended version or the problem at hand.

Yeah! Federico is our Team leader! he has shown us many times the proof of this beautiful result! anyway it is to hard to write it down, i just give some hints: Draw all circles and show that moving a point to another region which is a neighboor of it starting region doesnt change the number of halving circles! (the point may only pass trough one boundary) Conclude this number is invariant Use induction in $n$, to prove for $2n+1$ points it is $n^2$.

if you don't have access to the american mathematical monthly, you can read the article "the number of halving circles" here: http://www.math.washington.edu/~federico it is available as a pdf file. .f.

orl wrote: 5 points are given in the plane. Any three of them are non-collinear. Any four are non-cyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be good. Let the number of good circles be $ n,$ find all possible values of $ n.$ I think that $ n=3,4,5$ (using convex hull prove least 3 circle good ,least 5 circle not good) Am I right? :

math10 wrote: orl wrote: 5 points are given in the plane. Any three of them are non-collinear. Any four are non-cyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be good. Let the number of good circles be $ n,$ find all possible values of $ n.$ I think that $ n = 3,4,5$ (using convex hull prove least 3 circle good ,least 5 circle not good) Am I right? : I had wrong. $ n=4$ (not equal 3 and 5).

If you're going to post a solution, post a solution. Posting answers (right or wrong) without any proof is just pointless filling of space (especially when someone else has already stated the answer to a more general version of the problem). Federico Ardila's webpage is now here: http://math.sfsu.edu/federico/

JBL wrote: If you're going to post a solution, post a solution. Posting answers (right or wrong) without any proof is just pointless filling of space (especially when someone else has already stated the answer to a more general version of the problem). Federico Ardila's webpage is now here: http://math.sfsu.edu/federico/ please check link?(I can't open it )

It works fine for me. The article itself is at http://math.sfsu.edu/federico/Articles/circles.pdf