Can eight circles be constructed from three circles? Given three sufficiently spaced circles in a plane, is it possible, using a straight edge and compass, to construct the eight circles that are tangent to all three given circles?
yes, one outside all of them, one inside all of them, three inside one of them, and three inside two of them. As the question was edited, previously the three circles were of the same size and their centroids were concentric. the general case of three circles in space of arbitrary size and position is not always possible.

four of these circles are of significance, the rest can be obtained by rotations. One circle that is tangential to the outside of the circles, on that is tangential to the inside of all the circles. these two cannot be rotated to obtain more permutations. if you construct a circle that is tangential to the inside of ONE of the circles and the outside of the other two circles it can be rotated in space to obtain two more. if you construct a circle that is tangential to the inside of TWO of the circles and tangential to the outside of one, this can be rotated to obtain the remaining two.
yes, one outside all of them, one inside all of them, three inside one of them, and three inside two of them. As the question was edited, previously the three circles were of the same size and their centroids were concentric. the general case of three circles in space of arbitrary size and position is not always possible.

four of these circles are of significance, the rest can be obtained by rotations. One circle that is tangential to the outside of the circles, on that is tangential to the inside of all the circles. these two cannot be rotated to obtain more permutations. if you construct a circle that is tangential to the inside of ONE of the circles and the outside of the other two circles it can be rotated in space to obtain two more. if you construct a circle that is tangential to the inside of TWO of the circles and tangential to the outside of one, this can be rotated to obtain the remaining two.