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Lets say we have two circles with radii 2 and 3 respectively.

If we then put Circle radius 2 on a flat surface, then the other circle on the flat surface so they touch once, what is the distance between the two centers?

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up vote 1 down vote accepted

A the point of contact the circles share a mutual external tangent and so the two radii are both perpendicular to that line. Therefore the center to center distance is simply $r_1 + r_2$, in this case $2 + 3 = 5$.

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Clearly the question had been a homework question... – Sasha Aug 25 '12 at 18:40
but that tangent may not be perpendicular to the surface – fosho Aug 25 '12 at 18:41
@fosho What do you mean? Since the circles lie on the same flat surface I assume that they are embedded in the same plane. In that case any tangent to the circle also lies in the same plane containing the circles. – EuYu Aug 25 '12 at 18:55

If by "circle" you mean the set of points in a plane equidistant from a given point (the center), then there are two possibilities. If one circle is inside the other, then you'll subtract the smaller radius from the larger, and otherwise you'll add the radii.

If by "circle" you mean the set of points in a plane equidistant from a given point (the center) and all points inside it, then we can't put one circle inside the other and have them touch only once, so we must add the two radii. (This latter type of "circle" is usually called a "disk" instead.)

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Depends on whether you put the smaller circle inside or outside. In either case, the common tangent being perpendicular to both the radii at the tangency point would imply that the centers are collinear with this point. Hence the distance is either 5 or 1.

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