1. ## 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]

2. ## 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.

3. ## Re: A circle

Originally Posted by ChipB
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?

4. ## 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).

5. ## Re: A circle

Originally Posted by policer
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]
Originally Posted by ChipB
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).
Originally Posted by greg1313
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 $\ell$