Explorations with Tangent Circles


This discussion is a summary of my explorations with the basic idea of finding a circle tangent to two given circles. Sounds simple enough right? Brother, you don't know the half of it...


I am going to discuss this from a student's discovery standpoint, as I myself hadn't done anything like this before. I'm hoping to use my own experiences in working with this problem to help me understand how something like this could be used in the classroom. Here we go!


The first part of the assignment is to construct a circle tangent to both given circles who are contained within each other.

The construction of this circle comes from using isosceles triangles. The idea is to locate the center of the third circle. We know that the center of the third circle has to lie on the line containing the center of the large circle and the designated point of tangency on this circle.

We also know that the circles center has to be equidistant from the point of tangency on the large circle and a tangency point on the smaller circle. To find this, we are going to construct an isosceles triangle and find the perpendicular bisector of its base. We know from high school geometry that the perpendicular bisector of an isosceles triangle is the collection of all points equidistant from each endpoint of the segment. Finding where the perpendicular bisector intersects the line through the point of tangency will give us the location of the circle for that particular location.

In order to make our triangle isosceles we will construct a circle whose radius is equal to our smaller circle and center it on the point of tangency on the large circle. We find where this circle intersects the diameter of the circle containing the tangency point, and use this point and the center of the original smaller circle to form the base of our triangle.


Then we construct the perpendicular bisector of this segment. Where it intersects the line is the center of our circle (A).


And, we have our circle.

Having been given the information to accomplish this construction, we were asked to construct the purple circle such that the smaller circle is inside of it. Here is where the exploring comes into play. We need the tangent point on the small circle to be on the far side of it. How can we move our circle the equivalent of one diameter of the small circle? Hmm...

We know the basic idea is to form an isosceles triangle, but this time, we need the base of our triangle to end not at the far edge of the circle, but on the closer intersection point. This will accomplish our "one diameter closer" idea.

Now when we find the perpendicular bisector, and its intersection, we should have exactly what we need.

But constructing these circles is only the first part of the exploration. More interesting is to look at the collection of all of these circles. We have found the center of one of the circles, but by moving the point of tangency on the large circle, we can see all of the possible locations of the center of the purple circle. This collection of points is known as the locus of points for the center. We can use GSP to show the path the center takes.

If we look at the locus of points where the purple circle is tangent between the two green circles, we find this path, colored blue.

We can see that the path formed is an ellipse. The foci of the ellipse appears to be the centers of the two circles. Interesting...

If we look at the locus for the case where the purple circle is around the smaller circle, we see the following.

Once again the locus of points forms an ellipse, also appearing to have the center of the circles as its foci. You may notice that the first ellipse is more round than the second. This measure of roundness, or more correctly ovalness, is known as the eccentricity of the ellipse. It is determined by the distance from the center of the ellipse to one of the foci (c), and the distance from the center to one of the vertices (a) . The ratio of c / a is always between 0 and 1. The closer the ratio is to 0, the more circular the ellipse is, the closer the ratio is to 1, the more elongated the ellipse is.

In both cases the eccentricity is a function not only of the size of the circles, but also their location within the original circle. For the case where the smaller circle is contained within the purple circle, the locus of points will remain an ellipse with foci at the centers of each green circle, as long as the two circles remain inside of the other and do not intersect. Once the green circles intersect, the ellipses are replaced by...

a hyperbola!

Notice here that the centers of the circles are still foci, however, for a hyperbola instead of the ellipse.

In the case where the purple circle is between the green circles, intersection does not change the general shape of the locus. In fact it isn't until the circles are completely disjoint that the locus becomes hyperbolic.

Once again, we see that the centers of the circles are the foci for the hyperbola.

We see what happens when the circles are completely on the interior and when they are completely disjoint. what happens when the circles are tangent to each other?

It appears that as the circles are tangent, that the locus of points appears to collapse into a line segment connecting the two centers of the green circles.


Let's continue our exploration with these two circles. Now lets change the location of the smaller circle and see how it effects the shape of the locus.

Notice please, that as we have moved the circle farther away, the less severe the curve of the hyperbola is. As we get closer, the curve becomes more severe.


Now lets look at the second case, where the purple circle is constructed to encircle the smaller circle. The severity of the curve changes as the circles move farther apart. That is until, the circles are disjoint. At this point, the hyperbola does not appear to change.



However, once the circles were disjoint, I decided to start changing the size of the circles to see how that might effect the locus. I found something I thought was interesting.

The larger that circle C became, the closer the parts of the hyperbola became.

This continued to happen until...

The hyperbola collapsed into a line. I am hypothesizing that this is true when the circles are congruent. Which I show by measuring the radii of each circle above.

Then, I wanted to see what would happen if I moved the circles back towards each other until the circles again intersected.


The line remained! Notice also that it went though the points of intersection of the two circles!

What would happen if I made the two green circles concentric?

All three circles are congruent, and the locus of points for the purple circle is now only one point, the center of the two green circles. Cool!



I see this approach to the assignment as beneficial to a high school student in the following ways.

1. As a way to be introduced to the concept of locus of points.

2. As a means of reviewing constructions or the concept of tangency.

3. As a way of showing relationships between all of the different conic sections.

If any or all of these ideas are goals of a lesson, I recommend using a GSP exploration as a means to accomplish them.


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