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Thread: A circle

  1. #1
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    A circle

    A & B are points that given
    Can I draw more than two circle (with a given radius) at A & B? [The circles are with the same radius]
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  2. #2
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    Re: A circle

    In plane geometry (i.e. in 2-dimensions) there are either zero or two circles of a given radius R that can contain two given points (the zero case comes about if the points are separated by a distance greater than 2R). In 3-dimensional space two points don't define a plane, and hence there may be an infinite number of circles of a given radius that include the two points, all aligned on different planes, or again there may be zero such circles if the points are too far apart.
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    Re: A circle

    Quote Originally Posted by ChipB View Post
    In plane geometry (i.e. in 2-dimensions) there are either zero or two circles of a given radius R that can contain two given points ...
    In the plane if A = (0, 0) and B = (0, 2), then there is one circle of radius 1 that passes through both of those points at the same time.

    Are you referring to something else?
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    Re: A circle

    Good point, my bad. I was thinking of cases where 2R is less than or greater than the distance between the points A and B. For example if R = sqrt(2) then the center of a circle passing through (0,0) and (2,0) coud be either at (1, 1) or (1, -1). I should have thought about the case where 2R exactly equals the distance between A and B, which yields only one circle, in this case with its center at (1,0).
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  5. #5
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    Re: A circle

    Quote Originally Posted by policer View Post
    A & B are points that given
    Can I draw more than two circle (with a given radius) at A & B? [The circles are with the same radius]
    Quote Originally Posted by ChipB View Post
    In plane geometry (i.e. in 2-dimensions) there are either zero or two circles of a given radius R that can contain two given points (the zero case comes about if the points are separated by a distance greater than 2R).
    Quote Originally Posted by greg1313 View Post
    In the plane if A = (0, 0) and B = (0, 2), then there is one circle of radius 1 that passes through both of those points at the same time. Are you referring to something else?
    It surely clear that this question is about a plane (2-D). So lets say that $A~\&~B$ are two points.
    Then $d=D(A,B)>0$ the distance between the points is positive.

    Now I agree with Dennis the could have been a clearer posting. I am not clear about the given $R$ radius.

    Suppose that $\ell$ is the perpendicular bisector of the line segement $\overline{AB}$. Now any circle of radius $R$ that contains $A~\&~B$ has its center on $\ell$ If $R=0.5d$ there is only one center on $\ell$ If $R<0.5d$ there are no centers. If $R>0.5d$ then there are exactly two centers on $\ell$
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