You can explore the four circles below to manipulate where the points are on the circumference, drawing in all the line segments in the fourth circle, to see the maximum number of regions created for each set of points on the circle. Explore the pattern to see if it continues. How many examples are enough till you're convinced of a pattern? What constitutes a proof?

When there are 2 points on the circumference, how many maximum regions are created inside the circle?
3 points?
4 points?
5 points?
6 points?
Can you prove it?