Alex Szatkowski
Given two circles we can find two different tangent circles.
To begin, we need two circles, Circle A and Circle B.
Let P be a point on the circumferance of circle A. This will be the point of tangency of our new triangle. Next, construct a circle C with center P and the same radius as circle B. Construct a line k that passes through the center of both circle A and circle C.
Next, label the point of intersection of line k and circle C, I. From our new point I to the center of circle B create a line segment and label it j.
Next we will find the midpoint of line j and label it M and construct a line through M that is perpindicular to line j labeled line l. The point Q is where line l and k intersect.
Lastly, we will construct circle Q with point Q as the center and radius length QP. Circle Q is now tangent to the two given circles, circle A and circle B.
Here you will find a gsp tool for this tangent circle.
As I said before, there are two tangent circles. The other tangent circle contains circle B on the inside.
Let's look at when the given circles A and B have different characteristics.
1. When circle B is completely inside circle A
When circle B is completly inside circle A, the tangent (purple) circle is inside circle A as well.
2. When circle A and circle B overlap.
When the two given circles overlap, the tangent circle (color purple) in still inside circle A.
3. When the two given circles are disjoint.
When circle A and B are disjoint, circle A is enclosed in the tangent circle (purple)