Tangent Circles

By Venessa Smith

The goal of this assignment is to explore two circles and the circles that are tangent to both. When:

1) One circle is completely in the other cirle.

2) The two circles overlap

3) The two circles are disjointed.

Exploration 1: How many circles can be constructed that will be tangent to the two given circles.

This exploration revealed that in each case there were two distinct sets of circles that were tangent to both circles.

Demonstrations

Note: The orange and yellow circles represent the two distinct set of cirlces that are tangent to the two original green circles.

Click on animate buttons to see a dynamic demonstrations of the tangent cirlces

Case 1:

Case 2.

Case 3.

Exploration 2: Investigate the loci of the centers of the tangent circles.

Case 1.

Case 2.

Case 3.

Did you notice that the loci of the centers either traced ellipses or hyperbolas?

The centers form hyperbolas and ellipses because in each case when you add or subtract the distant between the center of the tangent cirlces to the centers of the original circles, the result is alway a constant. The constant is always the sum of the two radii of the original circles.

A cool animation: