c1 = 0 because there are no chords
i1 = 0 because there are no chords to intersect.
r1 = 1 because there is only 1 region bounded by an arc,
     (where the arc is the whole circle!)



c2 = 1 because there is 1 chord
i2 = 0 because there is only 1 chord, so no chords intersect.
r2 = 2 because there are two regions, 1 on each side of the chord.

 

c3 = 3 because there are 3 chords.
i3 = 0 because none of the chords intersect inside the circle.
r3 = 4 because there are 4 regions, a triangle, and 3 D-shaped 
     circle segments.

 

c4 = 6 because there are 6 chords.
i4 = 1 because there are two chords which intersect at 1 point
     inside the circle.  [We do not count where they intersect
     at the vertices, because they are ON the circle, not INSIDE.]
r4 = 8 because there are 4 regions, 4 triangles, and 4 D-shaped 
     circle segments.



c5 = 10, because there are 10 chords.
i5 = 5, because there are 5 intersection points inside the circle.
r5 = 16, because there are 10 triangles, one pentagon in the middle, and
         5 D-shaped circle segments around the edges of the big pentagon.



c6 = 15 because there are 15 chords.
i6 = 15 intersections because there are 15 intersections inside the circle.
r6 = 31 regions because there are 3 pentagons, 3 quadrilaterals,  
     19 triangles, and 6 D-shaped circle segments.





 

Edwin