**Assignment 7: Tangent Circles**

**Jason P. Pickhardt**

__Problem:__

Discuss the loci of the centers of tangent circles for all cases explored.

__Exploration:__

For the following exploration I explored three different cases of tangent circles:

1) One circle inside another.

2) Two circles separate from each other

3) Two circles that intersect.

These explorations were done using Geometer’s Sketch Pad and can be found here. Use the animation buttons provided to animate the point of tangency and observe the locus of the center of the tangent circle. It’s possible to resize the original circles to see how this affects the locus of the center of the tangent circle. It is important that you erase the original traces before doing so.

__Conclusion:__

When one circle is inside the other we have a case where the locus of the center of the tangent circle is an ellipse. Looking at several traces of this locus of points, we can make the conjecture that the ellipse constructed has foci that are the centers of the two original circles.

When the two original circles are separate from each other, we can see that the locus of points constructs a hyperbola. Again in this case the foci of the hyperbola are the centers of the two original circles.

The third and final case is similar to that of case one. The locus of points constructed by animating a tangent point forms a hyperbola with foci that are the centers of the two original circles. Also, notice that the locus of points goes through the points of incidence of the two original circles. When this happens there is no possible circle tangent to both original circles and there is simply a point of incidence.