Assignment 7: Tangent Circles

by

Melissa Silverman

Consider two circles, with one inside the other. How could you construct a circle tangent to both circles?

Begin by constructing a line through the center of the larger circle. Then construct a congruent circle (to the circle already inside the circle) whose center is where the line and the larger circle meet.

To find a circle tangent to the large purple circle and the blue circle inside of it, we need to construct a circle whose diameter is the length of the blue inscribed circle to the purple circle. From that we can conclude that the radius of this circle is half of that distance. Constructing a segment between the two blue circles and bisecting it will provide the base of the equilateral triangle we are looking for.

If we bisect this segment and mark the point of intersect between the perpendicular of this segment and our original green dashed line, we have the equilateral triangle that we need to construct a circle tangent to the purple and blue circles.

The point where the red and green dashed lines intersect is the center of a circle that is tangent to both the blue and purple circle.

The red circle is tangent to the inner blue circle and the purple circle. The red tangent circle can be moved around to illustrate many possible tangent circles. To use the tool and manipulate the construction yourself, CLICK HERE. In the pictures below, notice how the size and location of the red tangent circle differs, but the original two circles do not change.

Tracing the center of the tangent circle (red circle) creates an ellipse (the purple ellipse below). The two green segments represent the two lengths whose sum never changes as the center of the red circle moves around the ellipse. One green segment is the radius of the red circle. The other is the sum of the radii of the red circle and the solid blue circle.

In a middle school classroom, I think this construction would be hard for the students to understand conceptually. The notion of using an equilateral triangle to construct the tangent circle might leave the students confused. The construction is simple enough where students could follow a set of steps, but the reasoning behind the steps would most likely be lost.

As a teacher, I think that making the construction yourself and showing it to your students as an instructional tool would have many benefits. Students could sketch what they think the tangent circle to the two circles would look like on their own papers and compare their sketches to the GSP construction. By moving the tangent circle around, students can see that there are infinitely many ways to draw a circle tangent to the two circles. Animating the construction can be motivating and exciting for the students. Animating it after they have thought about and predicted the outcome would be a neat presentation.

In a high school classroom, this would be great for illustrating ellipses. This is a great demonstration of the definition of an ellipse. Animating the construction to form an ellipse helps students visualize exactly where an ellipse comes from. This is a far better method than just having students memorize the formula of an ellipse from the textbook and forcing them to accept why the formula is written the way it is and what the parts of the formula mean.