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Math Help - 3 circles incscribed in one

  1. #1
    Member realintegerz's Avatar
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    3 circles incscribed in one

    A circle of radius R inscribes three circles of radius T

    Find T in terms of R


    I have found that the three circles with radius T form an equilateral triangle



    The answer is supposed to be T=0.464R
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  2. #2
    Eater of Worlds
    galactus's Avatar
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    If we connect the vertices of that equilateral triangle into the center we have the radius point of the larger circle. Then, we create 3 isosceles triangles.

    Each of these triangles we can use the law of cosines.

    (2T)^{2}=a^{2}+b^{2}-2abcos(\frac{2\pi}{3})

    But a=b, so:

    4T^{2}=2a^{2}(1-cos(\frac{2\pi}{3}))

    a=\frac{\sqrt{2}T}{\sqrt{1-cos(\frac{2\pi}{3})}}

    Then radius of the large circle is R, so we have:

    R=\frac{\sqrt{2}T}{\sqrt{1-cos(\frac{2\pi}{3})}}+T

    Solve for T and we have:

    T=(2\sqrt{3}-3)R\approx .464R
    Attached Thumbnails Attached Thumbnails 3 circles incscribed in one-circleincircle.gif  
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  3. #3
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    Quote Originally Posted by realintegerz View Post
    A circle of radius R inscribes three circles of radius T

    Find T in terms of R


    I have found that the three circles with radius T form an equilateral triangle



    The answer is supposed to be T=0.464R
    As the triangle (formed by the centres of the three inscribed circles) in the attachment is equilateral we have:

    Oc=\frac{2T}{\sqrt{3}}

    and so:

     <br />
R=Oa=T+Oc=T\left(1+\frac{2}{\sqrt{3}}\right)<br />
    Attached Thumbnails Attached Thumbnails 3 circles incscribed in one-gash.jpg  
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