When I began this investigation, I wasnÕt sure exactly what to expect. This idea of creating tangent circles was new to me, so it took a little while for me to get used to what was going on.
After looking at the construction of the first type of tangent circle, I was amazed at how easily the other types followed. If you notice, when you do the construction of the first type of tangent circle, there are really two different points that there are to work with. We have a line that goes through our given point and the center of our larger circle. we create a circle with the radius of the smaller circle around the given point. If you notice, we have two points where this smaller circle intersects the line. For our first construction, we choose the intersection point that is outside the larger circles in order to create our first tangent circle, but there is no reason that we couldnÕt choose the other point. Choosing the other point gives us a tangent circle as well, only one that is interior to the larger circle and contains the smaller circle.
Once I had this concept down, the other tangent constructions were basically using this same technique. This type of exploration is one that I think is within the grasp of most students. The one thing I worry about might be whether or not most students will think of the first construction on their own. I had to think long and hard about what was going on and what I wanted to have happen. Once I made that first big leap, the others followed pretty easily.
One of the problems I think I might have had is a weakness in circles and geometry. Even though I have taken geometry in high school and am taking geometry this semester, I donÕt feel as though I have done very much in the way of circles. Most of the things that I have constructed dealt with lines and objects that we can create from them. I donÕt feel I knew enough about circles in order to be completely comfortable when I began this investigation, so this was one of those that I learned quite a bit from.