What is a tangent circle? How is one constructed? These two questions will be answered in the discussion below. For starters though, a tangent circle is a circle that is tangent to another circle at one point. The simple set of tangent circles occur when you only have two circles. This is shown in Figure 1 below.
The red circle is tangent to the blue circle in the above figure at point r. This is a basic description of a tangent circle. In this discussion, the tangent circles that will be discussed will be those that are tangent to two other circles at the same time. The following is a step by step walk through of how to construct a tangent circle that is tangent to two other circles at the same time.
With the construction of the tangent circle complete, now it is time to look a little closer at the tangent circles themselves. As point R moves around circle c1, what type of path would the center of the tangent circle trace out? A circle? A hyperbola? An ellipse? Figure 2 illustrates the path of the center of circle c3.
As you can see from this figure, the loci of the centers of the tangent circles(thick red line) trace out an ellipse. Why is it that the centers of the tangent circles trace out an ellipse? The answer to lies in the construction of the tangent circle. From the figure, where do the foci of the ellipse look to be? One may guess that the foci are located at the centers of circles c1 and c2. This guess would be correct. To understand this, look again at the figure associated with step 6 above. The distance from c1 to R is the radius of circle c1(in blue) and the radius of c2 is the same length as from points R to P because that is how we defined it in step three. Now looking again at the figure from step 6, the distance from point c3 to c2 is equal to the distance from c3 to P. This is true because point c3 was defined from a perpendicular line through the midpoint of the segment between c2 and P. Therefore, the distance from c3 to c2 plus the distance from c3 to c1 is the same as the sum the radius's of circles c1 and c2. This is the defintion for the foci of an ellipse.
What would you think the path of the loci of the centers of the tangent circles would trace out if circles c1 and c2 intersected? Figure 3 shows this occurence.
Once again, an ellipse is formed. However, take a look at Figure 4. Circles c1 and c2 still intersect but this time the loci of the centers of the tangent circles forms a hyperbola. This occurs because the tangent circle is tangent to the outside of circles c1 and c2 at all points other than the area that the two circles intersect. In this area the tangent circle is tangent to the inside of both circles.
There are many other investigations to do with tangent circles and if you would like to explore some of your ideas go to GSP, a tangent circle is already drawn and you can explore until your heart is content.