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.)