I found it particularly interesting to trace
the centers of the tangent circles as the point of tangency is
varied. When one of the given circles lies on the interior of
the second circle, the loci of the centers of the tangent circles
form two ellipses. **(Click here
to investigate.)** When the two given circles intersect,
the locus of the centers of one tangent circle forms a hyperbola
while the other locus forms an ellipse. **(Click
here to investigate.)** When one given circle lies on the
exterior of the other given circle, the loci form two hyperbolas.
**(Click here to investigate.)**