Assignment 7: Reflecting on Tangent Circles

By Krista Floer

The original problem that I looked at was constructing a tangent circle given a larger circle with a smaller circle inside. While looking at different cases, I also looked at the locus of one of the points in each case.

I first created a GSP script for tangent circles given a larger circle and a smaller circle inside that circle. I was able to construct the tangent circle for both these circles. After playing around with the tool for a bit, I discovered that it would construct the tangent circle in other cases. It works when the smaller circle is outside the larger circle and not touching it and also when the smaller circle intersects the larger circle. Click HERE for the GSP file containing the tool for constructing tangent circles.

I then started playing with the loci of the center of the smaller circle as the tangent point on the smaller circle moves. I noticed when the smaller circle is outside and not touching the large circle, the locus is a hyperbola.

As the smaller circle approaches the larger circle, the hyperbola keeps flattening more and more. I was not able to get an accurate picture of this using GSP. It seems that when the two circles become tangent to each other, the locus then becomes a line. Because the shape of the locus changes from a hyperbola to an ellipse, then I thought the locus would be a line at the point where the locus changes from a hyperbola to an ellipse. Then as the smaller circle approaches becoming completely inside the larger circle, the locus turns into an ellipse.

The locus keeps rounding out until the center of the small circle and large circle are concurrent. At that point, the locus becomes a circle.

Another thing that is interesting to note is the shape of the tangent circle. The case I was looking at was the disjoint case. When the point of tangency on the large circle is opposite the side of the smaller circle, the tangent circle is very close to the large circle on that side.

As the point is moved closer to the smaller circle, the tangent circle looks like it becomes a straight line. This would mean that the tangent circle turns into the tangent line at some point time. In the GSP file, if i moved the point upwards just a little, the circle disappeared. I think that this is the point were the circle turns into a line.

After exploring with the tool, I discovered I had found the basis and key concepts of answers to questions that were in the original problem set for this assignment. I think it is important for a student to explore math before they are given questions. For myself reading through the problems, I was wondering if I would be able to find answers and GSP sketches for most of them. I was just playing around and I found answers to those questions. The answers may not be completely thorough, but it gave me a jumping off point to start exploring my answers more deeply. From the teacher's perspective, I would rather begin with giving my students this construction and having them explore with it than have them get stuck in the mathematics of making the construction itself. I would love for my students to be able to make the construction in the first place, but if I had to choose I would rather them explore deeper questions than get stuck on the surface.

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