Question 1038060
<pre><b>  
{{{drawing(100,100,-5.2,5.2,-5.2,5.2, circle(0,0,5),circle(3,4,.4),
circle(3,4,.3),circle(3,4,.2),circle(3,4,.1)

)}}}
c<sub>1</sub> = 0 because there are no chords
i<sub>1</sub> = 0 because there are no chords to intersect.
r<sub>1</sub> = 1 because there is only 1 region bounded by an arc,
     (where the arc is the whole circle!)

{{{drawing(100,100,-5.2,5.2,-5.2,5.2, circle(0,0,5),line(3,4,4,-3))}}}

c<sub>2</sub> = 1 because there is 1 chord
i<sub>2</sub> = 0 because there is only 1 chord, so no chords intersect.
r<sub>2</sub> = 2 because there are two regions, 1 on each side of the chord.

{{{drawing(100,100,-5.2,5.2,-5.2,5.2, circle(0,0,5),line(3,4,4,-3),
line(3,4,-4,3),line(4,-3,-4,3) 
)}}} 

c<sub>3</sub> = 3 because there are 3 chords.
i<sub>3</sub> = 0 because none of the chords intersect inside the circle.
r<sub>3</sub> = 4 because there are 4 regions, a triangle, and 3 D-shaped 
     circle segments.

{{{drawing(100,100,-5.2,5.2,-5.2,5.2, circle(0,0,5),line(3,4,4,-3),
line(3,4,-4,3),line(3,4,-4,-3),line(4,-3,-4,3),line(4,-3,-4,-3), line(-4,3,-4,-3) 
)}}} 

c<sub>4</sub> = 6 because there are 6 chords.
i<sub>4</sub> = 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.]
r<sub>4</sub> = 8 because there are 4 regions, 4 triangles, and 4 D-shaped 
     circle segments.

{{{drawing(100,100,-5.2,5.2,-5.2,5.2, circle(0,0,5), 

line(0,5,-4.75528258,1.54508497),
line(-4.75528258,1.54508497,-2.93892626,-4.04508497),
line(-2.93892626,-4.04508497,2.93892626,-4.04508497),
line(2.93892626,-4.04508497,4.75528258,1.54508497),
line(4.75528258,1.54508497,0,5),
line(0,5,-4.75528258,1.54508497),line(0,5,-2.93892626,-4.04508497),line(0,5,2.93892626,-4.04508497),line(0,5,4.75528258,1.54508497),line(-4.75528258,1.54508497,0,5),line(-4.75528258,1.54508497,-2.93892626,-4.04508497),line(-4.75528258,1.54508497,2.93892626,-4.04508497),line(-4.75528258,1.54508497,4.75528258,1.54508497),line(-2.93892626,-4.04508497,0,5),line(-2.93892626,-4.04508497,-4.75528258,1.54508497),line(-2.93892626,-4.04508497,2.93892626,-4.04508497),line(-2.93892626,-4.04508497,4.75528258,1.54508497),line(2.93892626,-4.04508497,0,5),line(2.93892626,-4.04508497,-4.75528258,1.54508497),line(2.93892626,-4.04508497,-2.93892626,-4.04508497),line(2.93892626,-4.04508497,4.75528258,1.54508497),line(4.75528258,1.54508497,0,5),line(4.75528258,1.54508497,-4.75528258,1.54508497),line(4.75528258,1.54508497,-2.93892626,-4.04508497),line(4.75528258,1.54508497,2.93892626,-4.04508497) )}}}

c<sub>5</sub> = 10, because there are 10 chords.
i<sub>5</sub> = 5, because there are 5 intersection points inside the circle.
r<sub>5</sub> = 16, because there are 10 triangles, one pentagon in the middle, and
         5 D-shaped circle segments around the edges of the big pentagon.

{{{drawing(100,100,-5.2,5.2,-5.2,5.2, circle(-.1,0,5.2),

line(5,0,2.5,4.33012702),
line(2.5,4.33012702,-2.5,4.33012702),
line(-2.5,4.33012702,-4.635919273,1.873032967),
line(-4.635919273,1.873032967,-2.5,-4.33012702),
line(-2.5,-4.33012702,2.5,-4.33012702),
line(2.5,-4.33012702,5,0),
line(5,0,2.5,4.33012702),line(5,0,-2.5,4.33012702),line(5,0,-4.635919273,1.873032967),line(5,0,-2.5,-4.33012702),line(5,0,2.5,-4.33012702),line(2.5,4.33012702,5,0),line(2.5,4.33012702,-2.5,4.33012702),line(2.5,4.33012702,-4.635919273,1.873032967),line(2.5,4.33012702,-2.5,-4.33012702),line(2.5,4.33012702,2.5,-4.33012702),line(-2.5,4.33012702,5,0),line(-2.5,4.33012702,2.5,4.33012702),line(-2.5,4.33012702,-4.635919273,1.873032967),line(-2.5,4.33012702,-2.5,-4.33012702),line(-2.5,4.33012702,2.5,-4.33012702),line(-4.635919273,1.873032967,5,0),line(-4.635919273,1.873032967,2.5,4.33012702),line(-4.635919273,1.873032967,-2.5,4.33012702),line(-4.635919273,1.873032967,-2.5,-4.33012702),line(-4.635919273,1.873032967,2.5,-4.33012702),line(-2.5,-4.33012702,5,0),line(-2.5,-4.33012702,2.5,4.33012702),line(-2.5,-4.33012702,-2.5,4.33012702),line(-2.5,-4.33012702,-4.635919273,1.873032967),line(-2.5,-4.33012702,2.5,-4.33012702),line(2.5,-4.33012702,5,0),line(2.5,-4.33012702,2.5,4.33012702),line(2.5,-4.33012702,-2.5,4.33012702),line(2.5,-4.33012702,-4.635919273,1.873032967),line(2.5,-4.33012702,-2.5,-4.33012702) )}}}

c<sub>6</sub> = 15 because there are 15 chords.
i<sub>6</sub> = 15 intersections because there are 15 intersections inside the circle.
r<sub>6</sub> = 31 regions because there are 3 pentagons, 3 quadrilaterals,  
     19 triangles, and 6 D-shaped circle segments.

{{{c[n]=(matrix(2,1,n,2))}}}

{{{i[n]=(matrix(2,1,n,4))}}}

{{{r[n]=sum((matrix(2,1,n-1,k)),k=0,matrix(1,3,"","",4))}}} 

Edwin</pre>