 This is the write-up for Assignment 7

Exploring Tangent Circles

By

Doug Westmoreland

This exploration begins with the following problem:

Given two circles and a point on one of the circles, construct a circle tangent to the two circles with one point tangency being the designated point.

Here are some examples of the problem. The red circle is the tangent circle to the given two circles. The first three examples are when the one of the given circles is inside the other given one.   The next three examples show the the tangent circle when the two given circles intersect each other.   And, the final three examples show one of the given circles outside the other given circle.   For a dynamic presentation of the tangent circle click on one of the following below:

We need to also consider that the tangent circle could be drawn like the figures below.   For a dynamic presentation of the tangent circle in the above figures click here.

Now, let's take a look at some interesting discoveries that I found while playing around with the locus of the center of the tangent circle.

First, if the center of the two circles coincide, then the locus of the center of the tangent circle to these two will be a circle. See drawing below. Next, I noticed as the smaller circle (while remaining inside the larger cirlce) moved away from the center of the larger circle, the locus of the center of the tangent circle began to take the shape of an ellipse. See the drawing below. As the smaller circle continues to move away from the center of the larger circle, we see the locus of the center of the tangent circle getting more elongated. Finally, when the distance of the two centers is greater than the sum of the two radii the locus of the center of the tangent circle appears to take the shape of a hyperbola. 