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Math Help - Nested Tangent Circles

  1. #1
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    Nested Tangent Circles

    I need help with the following problem.

    All circles are tangent to each other, if the radius of the small circle is r then what is the radius of the largest one in terms of r? I can visually see that it is 6r, but I want to know why.

    I've tried constructing several triangles in the picture, but nothing clicks.

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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Nested Tangent Circles

    If we let x be the distance from the center of the largest circle (radius R) to the center of the smallest circle (radius r), and observing that the medium sized circle has radius \frac{R}{2}, we may use the Pythagorean theorem to state:

    \left(\frac{R}{2} \right)^2+x^2=\left(\frac{R}{2}+r \right)^2

    and we see that:

    R=x+r=\sqrt{r(r+R)}+r

    (R-r)^2=r(r+R)

    R^2-2rR+r^2=r^2+rR

    R^2=3rR

    For R\ne0, we find:

    R=3r
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    Re: Nested Tangent Circles

    Hello! Thanks for your reply!

    However, seeing as my drawing is not perfect I don't see how it is safe to assume that the center of the medium circle, the center of the smaller circle and their point of tangency lie on the same line. Am I missing something simple?
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    MHF Contributor MarkFL's Avatar
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    Re: Nested Tangent Circles

    The line tangent to both circles would be perpendicular to both radii, and so they both must have the same slope.
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    Re: Nested Tangent Circles

    Oh! I'm such an idiot, thank you!
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