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Math Help - Radius of circle touching three given touching circles

  1. #1
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    Radius of circle touching three given touching circles

    How to find the radius of a circle touching three given circles which are touching each other pairwise externally ?
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    The result was known to Descartes: if the radii of the four circles ar r1,r2,r3,r4 then 1/r1^2+1/r2^2+1/r3^2+1/r4^2 = (1/2)(1/r1+1/r2+1/r3+1/r4)^2. However it is usually called Soddy's Theorem because of the splendid way Frederick Soddy published it in Nature (June 1936):

    The Kiss Precise
    by Frederick Soddy

    For pairs of lips to kiss maybe
    Involves no trigonometry.
    'Tis not so when four circles kiss
    Each one the other three.
    To bring this off the four must be
    As three in one or one in three.
    If one in three, beyond a doubt
    Each gets three kisses from without.
    If three in one, then is that one
    Thrice kissed internally.

    Four circles to the kissing come.
    The smaller are the benter.
    The bend is just the inverse of
    The distance form the center.
    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 the squares of all four bends
    Is half the square of their sum.

    To spy out spherical affairs
    An oscular surveyor
    Might find the task laborious,
    The sphere is much the gayer,
    And now besides the pair of pairs
    A fifth sphere in the kissing shares.
    Yet, signs and zero as before,
    For each to kiss the other four
    The square of the sum of all five bends
    Is thrice the sum of their squares.



    See http://mathworld.wolfram.com/Descart...leTheorem.html
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  3. #3
    vms
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    Thanks a lot. But where can I get the proof for this?
    Last edited by MathGuru; July 18th 2005 at 05:45 PM.
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  4. #4
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    Solved the problem

    I have worked out the solution using properties of triangle, conditional identities in a triangle and few big calculations. It takes a little more time to type the same. As soon as I get it typed, I will upload it.
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