Problems on Circles III: Apollonius's Problem

Given three circles, this Demonstration shows the circles that are tangent to them. There are 0, 2, 4 or 8 possible tangent circles, depending on the configuration. You can modify the centers or radii of the three circles to draw the tangent circles with dashed circumferences. If the three initial circles are mutually tangent, then only two tangent circles arise, known as the Soddy circles.