For this assignment we will look at tangent circles.
First, let's look at a circle tangent to two given circles (one circle is inside the other).
The tangent circle is in red. The construction of a tangent circle is done using the perpendicular bisector of the base of an isosceles triangle as shown below.
Here is a script tool for creating a tangent circle similar to the one above.
Tangent circle tool (This will open a GSP file).
Next, we will animate the tangent point on the larger circle and trace the center of the tangent circle. When we do this we get an ellipse with foci at the center of the given circles.
If we animate the line passing through the center of the tangent circle we get the picture below. This shows how paper folding can be used to create an ellipse.
Here are some other tangent circles that can be constructed (the tangent circles are in red).
There are many different investigations involving tangent circles. The investigation dealt with on this page created an ellipse. One of the things learned in this investigation is how to construct an ellipse using tangent circles. This investigation points out relationships between tangent circles and other objects, such as an ellipse and hyperbola that I would have never thought about. This provides a good activity that shows geometric constructions of figures that student normally see only on graphs. This activity also helps combine geometry and algebra material, two subjects that most students do not think have anything to do with each other.