Tangent Circles

Allyson Faircloth

For this activity, we will be exploring tangent circles.

The first thing we need to do is be able to construct tangent circles. If you are given two circles, you can find a third triangle which is tangent to the original two circles.  The steps for this construction are below:

1. Given Circle A and Circle B.

2. Place an arbitrary point P on Circle A.

3. Construct a Circle C at that point with the radius of the Circle B.

4. Construct a line L through the center of Circle A and the center of Circle C.

5. Construct a point I which is the intersection of that line with the circumference of  Circle C.

6. Construct a line segment from the center of Circle B to point I.

7. Find the midpoint M of that line segment.

8. Construct a line K through M which is perpendicular to the line segment.

9. Find the intersection T of line K and line L.

10. Construct Circle T using point T as the center and line segment TP  as the radius.

11. Circle T is now tangent to the both Circle A and Circle B.

Here is a link to a GSP tool for constructing the tangent circle T.

However, there is also another circle which can be tangent to the two original circles.  This tangent circle is found by using the other intersection of the circumference of Circle C with the line L.  The steps for this construction are below:

1. Given Circle A and Circle B.

2. Place an arbitrary point P on Circle A.

3. Construct a Circle C at that point with the radius of the Circle B.

4. Construct a line L through the center of Circle A and the center of Circle C.

5. Construct a point I' which is the intersection of that line with the circumference of Circle C.

6. Construct a line segment from the center of Circle B to point I'.

7. Find the midpoint M' of that line segment.

8. Construct a line K through M' which is perpendicular to that line segment.

9. Find the intersection T' of line K and line L.

10. Construct Circle T' using point T' as the center and line segment T'P as the radius.

11. Circle T' is now tangent to both Circle A and Circle B.

Here is a link to a GSP tool for constructing the tangent circle T'.

For investigating a the tangent circle for two given circles, it is convenient to have both cirles on the same construction.

Here is a link to a GSP tool for constructing both tangent circle T and tangent circle T' at the same time.

Let's take a look at the two tangent circles together.

The first thing I looked at was whether the positioning of the two given circles would cause the constructions of the tangent circles to not work.

When the two circes are tangent next to each other, the first tangent circle now becomes our Circle A.

When the two circles are tangent in a way such that Circle A is around Circle B, the second tangent circle now becomes our Circle A.

Lastly, let's look at when the radius of Circle A is the same as Circle B.

We can see that the construction still holds.  However, there are two other tangent circles that could also be constructed.  They are shown on the image below.