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Math Help - different circles

  1. #1
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    different circles

    If three different circles are drawn on a piece of paper, at most how many points can be common to all three?
    A) Non
    B) one
    C) two
    D) three
    E)six

    E is the correct answer

    I could get four...but I cant get six at all! So is that enough reason to choose E?
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  2. #2
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    Re: different circles

    Quote Originally Posted by usre123 View Post
    If three different circles are drawn on a piece of paper, at most how many points can be common to all three?
    A) Non
    B) one
    C) two
    D) three
    E)six

    E is the correct answer

    I could get four...but I cant get six at all! So is that enough reason to choose E?
    Having in mind that 2 different(non overlapping/different centers or diameters) circles can intersect in at most 2 points, it is hard to imagine that 3 different circles can have 6 points in common.

    With that in mind i would be very interested to see your 4-point-intersection construction.
    Last edited by ChessTal; July 28th 2013 at 07:37 AM.
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  3. #3
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    Re: different circles

    Hello, usre123!

    Are you sure of the wording of the problem?


    If three different circles are drawn on a piece of paper,
    at most how many points can be common to all three?

    . . (A) none . . (B) one . . (C) two . . (D) three . . (E) six

    (E) is the correct answer. . ??

    Three circles can have at most one point in common.
    Code:
                  o o o           x x x
              o           o   x           x
            o               o               x
           o               x o               x
    
          o               x   o               x
          o               x   o               x
          o               x   o               x
    
           o               x o               x
            o             * ♥ *             x
              o       *   o   x   *       x
                  o o o           x x x
                   *                 *
    
                  *                   *
                  *                   *
                  *                   *
    
                   *                 *
                    *               *
                      *           *
                          * * *
    Perhaps the intent of the problem was:
    . . "How many points can be common to any two circles?"
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  4. #4
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    Re: different circles

    no, its all three, I just checked again. I'll upload my drawing in a day. I got four.
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  5. #5
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    Re: different circles

    Quote Originally Posted by usre123 View Post
    no, its all three, I just checked again. I'll upload my drawing in a day. I got four.
    The question is clearly flawed or it was incorrectly posted. I also find the replies astonishing.

    There is an infinite collection of circles having the property that any pair will have a given set of two points in common. The centers of the circles in that collection all are on the perpendicular bisector of the line segment between the two points.

    Equally important is theorem: Any set of three noncollinear points determines a unique circle. Therefore three distinct circles cannot have more than two points in common.
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  6. #6
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    Re: different circles

    Hi,
    Plato is absolutely correct. Also Soroban is correct if he adds the condition that the centers of the 3 circles are non-collinear. The attached drawing may be helpful:

    different circles-mhfgeometry31.png
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