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.
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