Investigation Seven

Tangent Circles

Chip Meyer


Creating tangent circles poses an interesting question to students. Do they understand what is really involved in creating a circle that is tangent to two other circles? The initial steps seem easy, but the understanding of these steps that may give them some fits.

Let's begin with two circles, one with Center B placed inside the other with Center A. I want to create a circle that is tangent to Point C on my larger circle as well as tangent to any point on my smaller circle.


I must now create a circle of the same radius as my Circle B at Point C.

Now I need to have a line that travels from the Center A of my large circle through my new circle at Point C and determine the intersection of these two items. I will label this point as E.

Point E is very important in this investigation because without it, the true center of the tangent circle can not be found. Using Point E, I must now create a line segement that travels from Point E to Point B, the center of my original smaller circle.

I must now find the midpoint of the segment BE. After this midpoint is found, a perpendicular line must be created that will also pass through the segment AE. The intersection of the segment AE and the perpendicular line from BE, marked X, will provide the center of the tangent circle I am trying to create.

Now, if create a circle with radius XC, I will have a new circle that is tangent to the inside of Circle A and the outside of Circle B.

As seen above, I was able to create the circle I originally intended but not without some trials and tribulations. It is important for students to think about the importance of segment BE in this construction. Without it and its perpendicular, one would not be able to create the tangent circle. Another insteresting part of this construction is the isosoles triangle EXB that is created as well. This triangle, shown below in red, is generated in part due to the midpoint of EB and its perpendicular to segment AE.

Overall this is a very interesting investigation that high school geometry students might find useful in their studies.


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