Below are the steps in showing how to create two tangent circles within one larger circle.
We will first start with constructing a circle and a line through the circle.
Next, create a smaller circle anywhere inside the larger one.
Now, copy the smaller circle such that the center of the new circle as at the intersection point of the line and the larger circle.
Then, connect the center of the smaller circle that is inside the larger one to the intersection point of our line and the second smaller circle outside the larger circle.
Next, construct the perpendicular bisector of the segment made above.
Now mark the intersection of the perpendicular bisector with the original line. We now have an isosceles triangle (outlined in red).
We now have the center and radius of the circle that is tangent to the smaller circle and the larger one. The center is the intersection point of our perpendicular bisector and our original line while the radius is the segment from this center to the intersection point of our original line and the larger circle.
Below is another drawing with only the three circles. We can also animate our intersection point around the larger circle and see that our tangent circle (pink) is always tangent to both the larger and the smaller circle.