In this investigation, I explore tangent circles; that is, tangents circles to two original circles. The two original have any placement possible--one inside the other, the two intersecting, or both disjoint. I have created a GSP Script Tool for the tangent circles and will explain how I did so. Here is the tool.

First Step: When constructing the tool, I began with two original disjoint circles (green), although the orientation of the two original circles is irrelevant. I then constructed an arbitrary point on one of the original circles, and constructed a line through the origin of the point and the center of the circle on which the point lies.

Second Step: Now we construct another circle with center as the arbitrary point and radius congruent to that of circle two. Then construct the intersection of this new circle the line previously constructed. Make a segment from this point the the center of circle two.

Third Step: Now construct the perpendicular bisector of this segment. The intersection of the perpendicular line and the existing line through the center of circle one will be the center of the tangent circle. Construct the tangent circle (red) using this point and the arbitrary point on circle one.

Final Step: When finalizing the tool, we can hide the construction objects so that the original circles (green) and the tangent circle (red) are the items to see.

Now we may need to discuss why we chose to construct a tangent circle in this manner. When constructing the perpendicular bisector, there were two congruent segments created. As well, there is an isosceles triangle constructed with the base being the constructed segment and the altitude being the segment given by the midpoint (perpendicular bisector) and the intersection of the perpendicular bisector and the line through the center of circle one and the arbitrary point. Because the triangle is isosceles, the sides are congruent (black and purple). We wanted this conclusion because we know the radii (black) of the circle two and the other circle are congruent. You can see the isosceles triangle here.

**Exploration of Tangent Circles:**

Now we want to use the tool we have to make some explorations about tangent circles. We will look at different orientations of the two original circles and make note of their loci. In order to do so, we will trace different points--such as the midpoint of a radius of the tangent circle and the center of the tangent circle.

**Case One: Circle Two lies inside Circle One**

After tracing the center of the tangent circle, the locus of this point appears to be an ellipse. Also after different trials as to size and orientation of the inner circle, we can also see that as the center of the inner circle gets closer to the center of the outer circle, the ellipse becomes a circle. We can view this happening with a script tool in GSP.

But how do we know if this locus is really an ellipse? Well an ellipse is the locus of all the points such that r_{1} + r_{2} = 2a, where a is the semimajor axis of the ellipse. In our instance, r_{1} is from the center of circle one (focus) to the center of the tangent circle and r_{2} is from the center of circle two (focus) to the center of the tangent circle. Then, what we want is r_{1} + r_{2} to be fixed. We can see that happening here using the GSP script tool. You will also notice that when constructing the two original circles, when the two centers are very close together (almost congruent) then r_{1} will be almost equal to r_{2}. This conclusion may be obvious since all radii of a circle are congruent.

**Case Two: Circle One and Circle Two are disjoint**

After tracing the center of the tangent circle in this case, we find the locus seems to be a hyperbola. This locus can be seen here using GSP. In a similar manner, we want to justify that this locus is a hyperbola. A hyperbola is the locus points of the difference of r_{1} and r_{2} such that r_{1} - r_{2} = k, where k is constant. In this particular instance, r_{1} and r_{2} are treated in the same way (center to center, with centers of original circles as foci). We want to find that the difference (r_{2} - r_{1}) is constant. This constant difference is shown here with the GSP tool.

**Case Three: The two original circles intersect**

As in case one, when the two circle intersect the locus is also an ellipse. We can see this locus here. We know that for an ellipse, r_{1} + r_{2} is constant, and this relationship can be shown here.