In this investigation, I looked at variations of a tangent circle constructed from two circles. Investigation of several situations, and by tracing the center of the tangent circle, reveals that the locus of points from a figure.

The first situation that I began looking at was when the smaller circle was external to the tangent circle as compared to the smaller circle being internal to the tangent circle.

Click here to view a GSP animation of this situation.

From viewing both of the above situations, the figure formed by the loci is that of an ellipse.

The next situation investigated was when the two given circles intersected.

Click here to view a GSP animation of this situation.

If the tangent circle is internal to one circle and external to the other, then the figure formed by the loci of the center of the tangent circle is an ellipse.

When the tangent circle is external to the two circles or internal to both of the circles, the locus of points forms the figure known as a hyperbola.

The final situation is when the two circles are disjoint.

Click here to view a GSP animation of this situation.

If the tangent circle is internal to one circle and external to the other, then the loci forms a hyperbola. If the tangent circle is external or internal to both circles, then the loci forms a hyperbola here also.

Through my investigation, I have been unable to find a parabola formed by the locus of points from the center of the tangent circle. However, I do think that there is some limited case as was questioned in the assignment. Possibly, future investigations will reveal this case.