Written by Sunny Yoon

The investigation started with a simple problem:

Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with 1 point of tangency being the designated point.

First, let's do the construction.

What happens when you trace the Center of Tangent Circle while moving the Point of Tangency along Circle B?

Is it possible to form another tangent circle with the same given condition? Yes!

So far, we've investigated finding a tangent circle when one circle is inside the other one. What happens with other scenarios? What if two circles intersect with each other? Is it possible to find a tangent circle? If it is, what happens to the loci?

As I was moving the Point of Tangency along Circle A, as the tangent circle (or Point of Tangency) gets closer to Circle B, the tangent circle disappears from inside of Circle A and was inside of Circle B. That was a difference from the earlier examples.

What if you find the locus?

Let's think about a scenario when two circles do not intersect with each other at all. Here's a picture of constructing the tangent circle.

Here's a picture when you construct the locus.

Is it possible to construct a locus that is a parabola? What about a circle? Is there any other possible scenarios involving two circles? Let's explore...

I'm going to find the tangent circle and the locus when two circles share the same center. Here's the investigation.

Now what happens when two given circles are already tangent to each other?

I'm going to move the center of Circle B so Circle B is still tangent to Circle A, but no longer inside of Circle A.

Click here for the GSP file. Move point B along the line.

As Circle B is still tangent, but outside of Circle A, you will see how the Center of Tangent Circle will move closer to Center A until it merges together. Therefore the tangent circle didn't necessarily "disappeared", but it's congruent to Circle A.