Erin
Mueller

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.