Subject: Re: Circle cutting
Date: Mon, 12 May 1997 00:04:09 -0400
From: Alexander Bogomolny

Andy:

The question you asked is not at all simple. I'll give you a couple of formulas. See if you need more help. Proceed as follows:

1. First of all assume that no three of your chords pass through the same point.
2. Note that the problem admits a different wording. x points on a circle form a convex polygon. Disregarding x circular segments between its sides and the circle, the question is into how many regions a convex polygon is divided by its diagonals. If A(x) is the number you are after, and B(x) is the answer to the latter problem, then A(x) = B(x) + x.
3. The answer to the problem #2 is obtained by computing the total number of points and the total sum of all angles in thus obtained regions.
4. C(x) = x(x-1)(x-2)(x-3)/24 is the number of points of intersection of all diagonals (this is the hard part!) Let we have r₃ triangles, r₄ quadrilaterals, etc. Then 3*r₃ + 4*r₄ + ... = 4*C(x) + x(x-2).
5. Counting the total of all angles, we have (r₃ + 2*r₄ + 3*r₅) * Pi = C(x) * 2*Pi + (x-2)*Pi.
6. Subtracting #5 from #4 gives

   2*(r₃ + r₄ + ...) = 2*C(x) + (x-1)(x-2)

   which simplifies to

   B(x) = r₃ + r₄ + ... = (x-1)(x-2)(x^2 - 3x + 12)/24

7. Note that A(6) = 31 and not 32 as you have claimed. Although A(5) = 16 indeed.

Best regards,
Alexander Bogomolny

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