Circles, circles, circles...
by
Stacy Musgrave
This writeup will be an exploration of circles. In particular, given two circles, we'll explore the possibilities for finding a circle tangent to the two given.
I started naively by looking at the special case where the tangent circle has center on the line connecting the centers of the given circles, as shown below. This case was fairly straight forward, as you simply connect the centers of the circles with a line. Then find half the distance between the points on the purple circles you want to have as tangent points. This will be the radius of the tangent circle centered at the midpoint between the tangency points. Do this for the four cases shown below.
The purple circles were fixed, and the green circles are all possible tangent circles with centers on the line connecting the centers of the purple circles.
But to the observant viewer, it is easy to notice there are certainly more tangent circles to the purple ones than described by the 4 green circles above. In fact, there are infinitely any of them. We can explore the family of solutions through the following link, where we consider two fixed circles, one completely contained inside the other: Animation to See Family of Tangent Circles for Embedded Circles
The next natural question is what happens when the circles are not as in the above case. For example, what if the two circles overlap? We show a picture to see what a solution tangent circle could like when the circles intersect.
Click on this link to see a family of solutions for such a case.
Finally, what happens when the circles are disjoint and neither is contained in the interior of the other? See the family of solutions we've found here.
The really cool thing is that all of these animations were actually created with the same construction. It would be an interesting activity to walk students through this construction and show how moving one of the circles to different locations in the plane does NOT impact the validity of the construction. Another point of interest is that for the case where the circles are disjoint, the tangent circle reaches a degenerate case where the tangent "circle" has infinite radius, i.e. is a line. I might challenge students to take an arbitrary image from one of these sketchpad files and actually prove that the blue circle is tangent to the two green circles. This activity would reaffirm the construction that would be demonstrated in class and give students the much needed practice in writing proofs. (Note: I am of the opinion that proof writing is an important skill in that students should be able to develope coherent, logical arguments! Giving students a construction and then asking them to translate the pictures into algebra is a skill that will be asked of them in many math classes to come.)
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