Assignment 7

Tangent Circles

By

Ken Montgomery

In this study we wish to explore the relationships between circles that are tangent and in particular the Locus of a circle that is tangent to two other circles.

First we use two script tools (Outside Tangent Circle, and Inside Tangent Circle), which can be downloaded from Assignment 5, to construct tangent circles. In the first investigation, we use the Outside Tangent Circle tool to perform the construction in Figure 1.

Figure 1: Black circle, tangent to two red circles

In Figure 1, the black circle is tangent to the two red circles. Here the tangent circle is inside a larger circle and outside of the smaller circle. If the tangent circle is animated, such that it orbits the small circle, while we trace its center, we obtain the locus in Figure 2.

Figure 2: The locus of the black (tangent) circle’s center

We label the center of the tangent circle, A, the center of the big red circle B and the small red circle C. Next, we construct segments and . We can see that as the circle rotates, the sum of the two segments AB and AC is equal to the sum of the radii of the two red circles (Figure 3).

Figure 3: Comparison of circle radii and center distances

The locus of the center of the tangent circle is an ellipse, although this is not proven in general. Download OutsideCircle.gsp to explore this further.

In the second investigation, we use the Inside Tangent Circle tool to perform the construction in Figure 4.

Figure 4: Red circle, tangent to two black circles

In Figure 4, the red circle is tangent to the two black circles. Here the tangent circle is inside a larger circle and containing the smaller circle. If the tangent circle is animated, such that it orbits the small circle, while we trace its center, we obtain the locus in Figure 5.

Figure 5: The locus of the red (tangent) circle’s center

Again, we label the center of the tangent circle, A, the center of the big black circle B and the small black circle C. Next, we construct segments and . We can see that as the circle rotates, the sum of the two segments AB and AC is equal to the difference of the radii of the two black circles (Figure 6).

Figure 6: Comparison of circle radii and center distances

The locus of the center of the tangent circle is an ellipse, although this is not proven in general. Download InsideCircle.gsp to explore this further.

As an extension of the second investigation, if the small black circle is moved outside of the large black circle, such that the two are disjoint, but the red circle is still tangent to both of them, the locus of the red (tangent) circle’s center is a hyperbola as demonstrated in Figure 7.

Figure 7: Red circle tangent to  and

Note that the difference between the segments and is equal to the difference between the two circle’s radii, and .

A similar extension is made of the first investigation. If we move the small red circle outside of the large red circle, such that the two are disjoint and still mutually tangent to the black circle, we have the locus given in figure 8.

Figure 8: Black circle tangent to  and

Note, again that the difference between the segments and is equal to the difference between the two circle’s radii, and .