You should be able to achieve the following configurations for the tangent circles by moving c2 to the interior of c1, the exterior of c1, and so that it intersects c1:

For c2 inside of c1, the tangent circle will always be internally tangent to c1 and externally tangent to c2.

For c2 outside of c1, the tangent circle will always be externally tangent to both c1 and c2. Something interesting to notice is that depending on where we position point D (or reposition circle c2), the tangent circle might surround either c1 or c2 as illustrated below.


Formalize some reasoning as to how these two cases come about. Discussion.

For c2 intersecting c1, the tangent circle will be internally tangent to one of the circles and externally tangent to the other as illustrated below.