I looked at three different cases for finding tangent circles:
1. Find the tangent circle to two circles, the smaller circle located inside the bigger circle. See the diagram below.
The two original circles are green and the blue circle is the tangent circle I found using GSP. The dashed lines were used in he process of finding the tangent circle. By duplicating the original smaller circle, then creating an isoceles triangle, and then finding the perpendicular bisector of the base of the isoceles triangle, the tangent circle can be found.
To open GSP and use a script tool 'tangent circle' to
create your own example click on the picture above.
Really there is a family of tangent circles that can be found for these two circles. In order to visualize this idea click on the diagram below. GSP will open and then click on the Animate Point B button - watch what happens.
As Point B moves counter-clockwise around the larger original circle, the tangent circle appears to change size in order to remain tangent to both the larger and smaller circle. There are actually many tangent circles (a family) and by using the trace tool in GSP to trace the center point of the tangent circle, you can see that an ellipse is formed with the foci located at the center of the two original circles. See diagram below.
2. What if the two original circles were overlapping or intersecting? Could the tangent circle be found the same way as above? Check out the diagram below. The two original circles are red and the tangent circle is blue.
The answer is yes! The isoceles triangle, CDE, was drawn using the duplicate smaller circle (dashed line). Then the perpendicular bisector to the base of this triangle was drawn - this helped to find the center of the tangent circle.