I did understand that you were talking about counting the regions not the lines, so that's OK. You were right that I forgot the two crossing lines in my tic-tac-toe example, but I think that what I said there is still true: by changing the positions of the points, you can change the number of regions that the circle is cut into, therefore there is no unique answer to your question.

For 3 points, I agree that you must get 7 regions, but for 4 points, you must get 16, 17, or 18 regions depending on how your points are arranged. For example, if you take the circle to be x2 + y2 = 1 and let the 4 points be (x,y) = (-0.1,-0.1), (-0.1,0.1), (0.1,-0.1), (0.1,0.1), you will get a tic-tac-toe board with a cross through the center, which gives 16 regions. BUT, if you let the 4 points be (x,y) = (-0.2,-0.1), (-0.1,0.1), (0.2,-0.1), (0.1,0.1), the lines that were vertical in the original tic-tac-toe shape will now meet at (0,0.3), and a new region will appear above this point, making the total 17 regions. You can also distort the positions of the points so that the lines that were horizontal in the original tic-tac-toe shape will now meet inside the circle, so you can get 18 regions.

I am sorry that I forgot the two diagonal lines in my original tic-tac-toe example, as this made my point less clear. I think that if you draw what I described you will see that you get different numbers of regions depending on how the 4 points are arranged. Notice that no three of these points are collinear, so they still satisfy your original requirements.

Notice that this does not depend on the original lines being parallel, as they were in the tic-tac-toe example. If you have two pairs of points inside the circle, and they define lines that are nearly parallel, then the meeting point of these lines could lie outside the circle. Then moving the points slightly to change the angles of the lines could being this meeting point inside the circle, where it would cause one more region to appear.

That is why I said before that the circle gives trouble. If you just wanted to cut the whole plane into regions using the lines defined by n points, then none of this trouble could happen, because all intersection points would be included automatically, instead of some being inside the circle and others outside it. You would still have trouble with parallel lines, because they would cut the plane into fewer regions than non-parallel lines. In fact, the example I gave here with the point coordinates shows that the whole plane would be cut into 16 regions with the first set of points, and into 17 regions with the second set of points.

Therefore, to get a definite number, you have to disallow parallel lines. It is also possible for several lines to meet at a common point, which would reduce the number of regions the plane (or circle) was cut into. For example, it you added to my original 4 points in the example above, another two points at (x,y) = (0,-0.5), (0,0.5), they would generate several more lines with the other points, but one particular line would give trouble: the line joining the new pair of points will meet the original pair of diagonal lines at the origin, so that three lines are concurrent there. Now if I move the new pair of points a tiny bit to the right, so they are now (0.0001,-0.5), (0.0001, 0.5), the line joining them will miss the origin by a little bit, and there will be a tiny triangle contained between it and the two diagonals, which was not there before. None of the other regions will be affected by this (draw it to see), so the total number of regions will be increased by 1 when I move these points.

This means that to get a definite answer, you must require that none of the lines are concurrent, none are parallel, and there is no circle. In that case, your n points will generate n(n-1)/2 lines, and as I said before, the problem becomes one of finding how many regions the plane is cut into by n(n-1)/2 lines that are in general position (i.e., none parallel, none concurrent).

I hope this is more clear than my earlier post.

Good luck in solving this problem.

--Stuart Anderson