First let's construct the script for creating tangent circles.
The first thing you do is construct two circles, where one is inside the other but they are not touching. Then we will make a line through a random point on the outside circle and through the center of that circle.
Next we need to create a circle with the same radius of the smaller circle and center A of the outside circle. Then we need to create a point on the new circle where it intersects the part of the line outside the larger circle. We can label the intersection B.
Next we construct a line segment from the center of the inside circle to point B, inside the new circle. Then we find the midpoint of that segment and create its perpendicular bisector. The point where the perpendicular bisector hits the line that is inside the circle we will label C. C will be the center of our circle that will be tangent to both circles.
Then we will create our tangent circle with center C and radius to A. The circle will be tangent to both circles.
What is the locus of the tangent circle?
When there is a smaller circle inside a larger circle, the red trace is locus and it is in the shape of an ellipse with foci at the centers of the two original circles.
What happens when the circles are not inside each other or disjointed?
When they are disjointed, the circles don't form an ellipse, the locus is a hyperbola, instead. The foci are also the centers of the original circles as well.
What happens when the circles intersect each other?
When the circles intersect each other, they continue to form an ellipse and the foci of the ellipse are still centers of the original two circles.
What happens if the two circles are the same size and right on top of each other?
You can see that if the circles are right on top of each other, then the locus of the tangent circle is the actual circle itself.
To explore more check out the script tool.
By Carolyn Amos