First, we need to construct tangent circles and geometer’s sketchpad is the easiest way to do this.
1) Construct a circle with a smaller circle inside of it.
2) Then, we can copy the small inner circle and place it at the center point of tangency.
3) Make segments of the radii of each circle for later use.
4) Now, we can construct a circle by center and radius at the point of tangency and the smaller radius.
5) Construct a line and then a line segment from the center of the larger original circle to the center of the newest circle.
6) Next we can draw a line segment from the center of the small inner circle to outside point of the other circle. Then we can create the perpendicular bisector of this line segment.
7) From this, we can form a triangle. The third vertex created from this triangle will be the center of the circle tangent to both smaller circles.
We can also construct a third tangent circle through a similar construction. This time, we will extend a line through the other small circle’s diameter and construct the perpendicular bisector. This intersection will allow us to draw another triangle such that one side of the triangle will act as the center and radius of another tangent circle.