At first I was going to have my kids do the creating of the tangent circles and I would write up how they did and felt about the exploration.† But first I felt that I should try it so they could ask me questions as they went along.† Well my personal exploration put an end to that idea.† I found it very complicated to construct the tangent circles correctly.†I had one script created for a pair of circles but when I changed the radius of my smaller circle the radius of the tangent circle did not adjust also.† Thus the circles were no longer tangent.† Because I was running into more problems than expected, I decided to rediscover circles and their tangents after years of being away from it.
The point of tangency of two circles is the one point where the tangent circle touches each circle simultaniously with on over lap.†Itís almost as if they kiss at these points.† Here is an example of two circles having tangent points with another circle.† The red circle only touches the large blue circle at one point and the smaller blue circle at onepoint.† These would be the points of tangency.
If you would like to move the circles around yourself click here (Tancirc1.gsp). Now we will see if the same script will work when the circles are not one with in the other.
The script created does work for this situation also.† When animated the trace made by the rotating around the circle A is completely different from the trace made by the two circles one inside the other.† The circle contained in the circle forms a trace of an ellipse, while the circle next to the circle forms a hyperbola.†To see the animation and change the size of the circles click here.(tancirc2.gsp)
What if the circles over lap?† Will the script work for that one? And what shape will be formed when it is animated?
Personally I think the animation of this set of circles is the most interesting to watch.† It took me a couple of minutes to actually see the path it was taking.† When the tangent circle is animated to rotate around the larger circle it creates an elliptical path around the common area shared by the two circles.† To see the animation and change the size of the circles click here. (tancirc3.gsp)
This is a very involved topic to look at. This has been a short over view of the basics covering tangent circles.† As I discovered very quickly, this is for the more advanced math student and not kids in middle school.