So the intersection of the line through the center of the larger circle and the perpendicular bisector of the base of the isosceles triangle is a good candidate for the center of a tangent circle. So, let's draw it.
The dynamic power of GSP allows us to drag the tangency point on the larger circle around and see many different tangent circles. We can even animate this action and trace the locus of the center of the tangent circle. It gives us an ellipse as seen below.
For a script tool and the animation, click here.
Are these the only possible circles that are tangent to both of the original circles? Consider the following construction.
Here we have a tangent circle that contains the smaller circle. Again, we can animate the tangent point of the larger circle and trace the locus of the center of the tangent circle. What do you think this locus will look like? Click here for the script and script tool.
Can you create your own constructions where the known tangent point is on the smaller circle?
Click here and here for examples.
Now consider what would happen if the two circles intersected.
We can do the same type of construction as before.
The above picture shows the case where the tangent contains the smaller circle (initially). Can you do the other case?
What is the locus of the center of the tangent circle now?
Now it is a hyperbola. Did that surprise you? See if you can find another case. To explore this case more, click here.
What other cases could you try? What other arrangements could you make with the original circles besides these two?
Here is one more example: Disjoint 1
Did you come up with this or others?