Descartes Circle Theorem

Date: 01/15/2008 at 08:58:43
From: Don
Subject: Largest circle inscribed in 3 mutually tangential circles

What is the radius of the largest circle which can be inscribed 
within the area formed by three mutually-tangential circles, in terms 
of the radii of the three circles?

I can only get a formula by working out what is consistent with the 
results of simpler examples, for example three equal circles, two 
equal circles or two circles and a straight line (circle of infinite 
radius).   

If the radii of the three circles are a, b and c then the formula I 
have for the radius of the inscribed circle r is 1/r = 1/a + 1/b + 
1/c + 2sqrt[(a + b + c)/abc].  I would like a proof of whether it is 
correct or not.

Date: 01/15/2008 at 10:59:13
From: Doctor Tom
Subject: Re: Largest circle inscribed in 3 mutually tangential circles

Hi Don,

What you are looking for is called the Descartes Circle Theorem.

If the radii of four mutually-tangent circles are r1, r2, r3 and r4,
then if:

  a = 1/r1, b = 1/r2, c = 1/r3 and d = 1/r4 

the theorem states that:

  2(a^2 + b^2 + c^2 + d^2) = (a + b + c + d)^2

Notice that the formula above is quadratic, so there are two
solutions.  If you have 3 mutually-tangent circles, one of the
solutions will be for the circle that's "inside" the other three, and
the other will be for the exterior tangent circle.

Also, the formula works if one of the "circles" is a straight line, in
which case the "radius" is infinite, so 1/r = 0.  If a solution is
zero, it just means that there's a line that's tangent to the three
circles.  If there are two zero solutions, there are two tangent 
lines.

I haven't really researched this looking for good proofs, but maybe
you can find one somewhere on the Internet.  I just wrote down the
equations by brute force and plugged them into Maple (a computer
algebra program).  About 400 lines of calculation later Maple verified
that the formula above is correct.

You can also look up "Soddy circles" since they're basically the same
thing.  I'll bet you didn't expect a poetic answer, but I even have
one of those:

  Four circles to the kissing come,
  The smaller are the better.
  The bend is just the inverse of
  The distance from the centre.
  Though their intrigue left Euclid dumb
  There's now no need for rule of thumb.
  Since zero bend's a dead straight line
  And concave bends have minus sign,
  The sum of squares of all four bends
  Is half the square of their sum.

  -- Frederick Soddy 

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/17/2008 at 14:48:26
From: Don
Subject: Thank you (Largest circle inscribed in 3 mutually tangential
circles)

Many thanks for your reply to my question about three circles.  It
also answered another suspicion I had about the formula I sent to you
and that was that the square root term could be positive or negative.
I suspected that the negative square root would give the radius of the 
exterior tangent circle but had no way of verifying it.  I haven't yet
had a chance to look up the Descartes Circle Theorem on the internet
but it has given me much more of a lead than I have had in the past. 
I am really very impressed by the promptness of your reply and once
again, many thanks for your help.  All the best, Don.