I get 4 such circles (radii):
If the smaller circle is inside the larger, then the third circle can only be internally tangent to the larger and externally tangent to the smaller to form a triangle of centers (rather than a line of centers if it were internally tangent to both). It could be a tiny one near the point of tangency of the first two, or larger as it goes around to being in line with the other centers and have radius 10. In between, the center will pass a line that's perpendicular to the line of centers of the other two and going through each of the first two centers. So this accounts for two circles. (There are another two on the other side, but we are told to discount congruent triangles.)
If the smaller circle is externally tangent to the larger, then the third circle must be outside the two original circles (otherwise it's center would be in line with the other two centers). It can be such that the other two circles are internally tangent to it, or externally tangent to it. In the case the other two circles are internally tangent to the third, we can make the diameter of the third circle equal to the combined diameters of the first two, leaving the center in line with the other centers, or push the external circle off to one side, making it larger until it is at the point where its center to the center of the larger circle and then to the center of the smaller one is a right angle. This can be done on either side, but by symmetry the two results are the same (congruent, but opposite).
If the third circle is externally tangent to both original circles, it can be made large enough so its center makes a line with the center of the smaller circle that's perpendicular to the original line of centers.
Thus I count 4 such circles.
Posted by Charlie
on 2004-05-27 15:26:52