Exploring Tangent Circles
by
Rita Meyers

During this exploration we will look at a circle that is Tangent to two given circles.
We will look at this in several different ways.

The first case we will look at is if one of the given circles is inside the second given circle.

First we will explore what happens when the smaller given circle is external to the tangent circle.

In order to explore these circles let's discuss the locus of the center of the tangent circle.
Exploration #1

If we move the tangent circle along the large circle we see that the locus of the center of the tangent circle looks like this

and if we move the circle along the small circle we get this locus

In each case it appears that the locus is an ellipse with the centers of the given circles your foci; I'll go further to say that they are the same locus

What happens when the smaller circle is internal to the tangent circle.

Now, let's explore these circles by discussing the locus of the center of the tangent circle.
Exploration #2

Again if we move the tangent circle along the large circle we see that the locus of the center of the tangent circle looks like this

And if we move the tangent circle along the small circle we get the same locus; which again appears to be an ellipse with the centers of the given circles your foci.

What happens if our two given circles intercept (Use the same construction method as the given circle internal to the tangent circle)?

Let's again explore by discussing the locus of the center of the tangent circle.
Exploration #3

Once more we have a locus that looks like this.

If we move the tangent circle along either given circle we get a locus that once again appears to be an ellipse with the foci the centers of the given circles.

Lastly, we will look at what happens if our given circles are completely separate of each other (Again, use the same construction method as the given circle internal to the tangent circle).

We'll also explore this situation by looking at the locus of the center of the tangent circle
Exploration #4

Here we are a totally different locus that previously; we now have one that looks like this.

When we move the tangent circle along either given circle we get a locus that this time appears to be a hyperbola with an asymptote