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Thread: touching circles

  1. #1
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    touching circles

    I am looking at this image, wondering is it obvious that if these two circles touch then there must be a straight line that passes through the centre points?

    http://www.cut-the-knot.org/proofs/ford2.gif

    Is this based on a circles theorem? Can anyone help please?
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  2. #2
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    Re: touching circles

    Hey rodders.

    Try using standard trig [i.e. involving right angles] and basic identities involving circles.

    We don't do your work for you.
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  3. #3
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    Re: touching circles

    Thanks. Will give it go. Just wanted a hint.
    I am thinking the distance between the two centres is the sum of the radii ? Or thinking there will be one point of intersection between the two circles rather than two ? And do it algebraically? I will think
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  4. #4
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    Re: touching circles

    Yes, when two circles touch at a point, it is easy to show there is a straight line between their centers that passes through that point of intersection.
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  5. #5
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    Re: touching circles

    Is the straight line that passes through the two centres perpendicular to the line that is formed by subtracting the two equations from each other?
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  6. #6
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    Re: touching circles

    What two equations are you subtracting from each other?
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  7. #7
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    Re: touching circles

    Two circle equations.
    (x-4)^2 + ( (y-3)^2 =9
    and
    (x-a)^2 + (y-1)^2 = 1

    I am trying to find the value of a so the smaller circle touches the bigger one.
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  8. #8
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    Re: touching circles

    Quote Originally Posted by rodders View Post
    Two circle equations.
    (x-4)^2 + ( (y-3)^2 =9
    and
    (x-a)^2 + (y-1)^2 = 1

    I am trying to find the value of a so the smaller circle touches the bigger one.
    The distance between their centers will be $R + r = 3 + 1 = 4$

    large circle has center $(4,3)$, smaller circle has center $(a,1)$

    $(a-4)^2 + (1-3)^2 = 4^2$

    note there are two possible values for $a$ ...
    Attached Thumbnails Attached Thumbnails touching circles-circle_touch.jpg  
    Thanks from rodders
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  9. #9
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    Re: touching circles

    Thanks.
    I managed to work that out too. I am thinking about another circle inbetween the two touching both!
    Can't see a simple way of doing this!

    http://www.cut-the-knot.org/pythagor...onSangaku1.gif
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