Some of the other constructions are extremely complicated. You will see something similar to the picture at right. This is one case in which it is difficult to dispute the value of dynamic geometry software. With so many interdependent objects, I would never be able to complete this drawing using straightedge and compass. It would be too difficult to distinguish between the different objects, the precision would be compromised with each step, and so many construction marks could not be cleanly erased.
Circle inversion is the key.
It would seem difficult to find a circle that is tangent to three given circles. However, if two of those circles were concentric, then it would not be so difficult at all. Given any two non-intersecting circles, it is possible to define an inversion such that the circles' images are concentric. Invert all three circles, construct tangent circles, and run the solutions through the same transformation. Their images will be tangent to the original circles.