Assignment 7
Cases of Tangent Circles
by Debra Jackson
i. When one given circle lies completely inside the other
Some items of interest include:
1. The yellow triangle formed by connecting the centers of circles A, C, and F is an isosceles triangle.
2. When center of circle F is traced, it forms the conic section ellipse when the constructed circle A's center does not align with the center of circle B. Click here to try it out yourself.
3. If the center of the constructed circle F is connected by segments to the centers of the two given circles, it is immediate that the sum of the segments is the same as the sum of the radii of the two given circles. This the sum is a constant and therefore the locus of the centers of the tangent circles is an ellipse with foci at the centers of the given circles.
4. If the constucted circle A is moved to become concentric with constructed circle B, then the locus of points of the center of circle F becomes another concentric circle. Click here to try it out yourself.
ii. When the two given circles overlap
When the center of circle F is traced, the loci forms an ellipse. When the center of circle E is traced, the loci forms an hyperbola.
Click here to see for yourself!
iii When the two given circles are disjoint
and when the centesr of circles E and F are traced, the loci form 2 pairs of hyperbola.
Click here to see for yourself.
One of the observations from this exercise is that the constructed circle A remains tangent to both circles E and F no matter what relationship it has to constructed circle B, inside, overlapping or disjoint, when the variable point of tangency C is moved about constructed circle B.
If the midpoint G of the base of the isosceles triangle is traced it's locus is always a circle no matter what the location of constructed circle A in relationship to constructed circle B.
The other two circles G and F remain tangent to the constructed circles A and B not matter the location of Circle A in relation to circle B.
If the blue tangent circle is traced while constructed circle A is inside constructed circle B, the trace fills in the space between these two constructed circles. Click here to see for yourself.
If the red tangent circle is traced while constructed circle A is inside constructed circle B, the trace fills in the space between these two constructed circles. Click here to see for yourself.
Another trace that is interesting to look at is when the two perpendicular lines are traced.
Circle Inside 
Circle Overlapping 
Circle Disjoint  Try it yourself  
Red Line 
Click here and move small green circle to various locations  
Result  Ellipse 
Ellipse 
Hyperbola 

Blue  Click here and move small green circle to various locations  
Result  Ellipse 
Hyperbola 
2 Hyperbola 