# By Donna Greenwood

This assignment 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 of tangency being the designated (given) point. To see more about the assignment, click here.   To aid in the discussion, let one of the circles be larger than the other (no particular reason here, except that the terms 'big circle' and 'small circle' have meaning).  The small circle can then be internally or externally tangent to the constructed circle when the small circle is on the large circle's interior.  The given point of tangency can lie either on the big or small circle.  That gives four cases to examine, but the construction of the tangent circle is essentially the same for all four.  If you're interested in a particular case, use the links to get there directly.
Case 1: Pink Circle:  The given point is on the big circle, and the small circle is internallytangent to the constructed circle when the small circle is in the interior of the big circle.
Case 2: Green Circle:  The given point is on the big circle, and the constructed tangent circle is always internally tangent to one of the given circles and externally tangent to the other.
Case 3: Red Circle:  The given point is on the small circle, and the constructed tangent circle is always internally tangent to one of the given circles and externally tangent to the other.

Case 4: Purple Circle:The given point is on the small circle, and the small circle is internally tangent to the constructed circle when the small circle is in the interior of the big circle.

## Construction Description

This picture shows the construction of the tangent circles when the given point is on the large circle.  The blue circles are the givens.  The pink circle is externally tangent to both given circles, and the green circle is internally tangent to the big circle and externally tangent to the small circle.  Click here for a sketch with the construction lines left in. To construct draw a line through the given point and the center of the circle on which the given point lies (call this one circle A, and the other given circle B). This is the aqua line in this picture. At the given point, construct a circle with circle B's radius (call this one circle C). To get the pink circle, construct a segment from the 'far' intersection of circle C with the line to circle B's center. To get the green circle, draw a segment from the 'near' intersection of circle C with the line to circle B's center. These are shown as purple segments in the sketch. In either case, get the midpoint of the segment and construct a perpendicular at that point. The perpendiculars are shown in orange. The intersection of the perpendicular with the original line is the center of the tangent circle.

If the given point is on the small circle, interchange the circle A's and circle B's in the description above, and the construction holds. For this scenario, the red circle mimics the green circle above, and the purple mimics the pink circle. Click here to see the construction

The center of the tangent circle traces out some interesting patterns.  How do these patterns compare among the different 'flavors' of tangent circles described here?

## Case 1: (Pink Circle)

The given point is on the big circle, and the small circle is internally tangent to the pink circle. The blue circles are the given ones, and the pink circle is the tangent circle. Note: the pink circle is the same one described in the construction description. Click here for a sketch to manipulate.

The locus of the center of the tangent circle traces out an ellipse for this case when the small circle is inside the big circle. As the centers of the two given circles become closer, the trace approaches a circle.

When the two given circles are disjoint or intersect, the center of the tangent circle traces a hyperbola. As the two blue circles get closer together, the hyperbola flattens out. This flattening continues until the small circle is almost inside the big circle:

When the small circle is close to being internally tangent, the pink circle 'disappears' (it's actually coincident with the big circle) except for an instant as the point is rotated around the big circle. The center of the tangent circle traces out an ellipse so flat that it looks like a line.

## Case 2: Green Circle

What about the green circle in the construction description? In this case, the tangent circle is always internally tangent to one of the given circles and externally tangent to the other. Do the loci behave differently for that version of the tangent circle? Click here if you want to see this version of the sketch.

When the given circles are disjoint, the center traces out the hyperbola. This looks similar to the case with the pink circle.

When the two given circles are externally tangent, the center of the tangent circle traces out the flat ellipse seen here.

When the given circles intersect, the center of the tangent circle traces out an ellipse, not the hyperbola seen with the pink circle.

The green circle also traces out this ellipse when the small given circle in on the interior of the big circle.

## Case 3: Red Circle

In this scenario, the two blue circles are the given circles, and the given point is on the small circle. One of the blue circles is always internally tangent to the red circle, and one is always externally tangent. Because of this, we should expect the traces to look the same as for the green circle above. Click here for this sketch.

When the circles are disjoint, the familiar hyperbola is traced out. The hyperbola gets flatter as the centers of the two given circles approach each other.

When the two blue circles are externally tangent, the flat ellipse is traced out.

As with the green circle, when the two circles intersect or when the small circle is on the interior of the big circle, an ellipse is traced out.

## Case 4: Purple Circle

Here's the purple circle. Check out the sketch to make sure that you can validate that the purple circle center traces out the same patterns that the pink circle's center did.

## Final Note:

One interesting thing that falls out of analyzing the different 'flavors' of tangent circles is that regardless of the type of tangent circle that is constructed, the center of that tangent circle traces out a hyperbola if the two given circles are disjoint. That says that the hyperbola is not dependent on the tangent circle(s). To understand why that's the case, look at the definition of a hyperbola:
Each hyperbola is a curve on a plane that is determined by two different points, F1 and F2, called foci (the singular is focus) and a real number c, with 0 < c < d(F1, F2). It consists of all points P such that
|d(P, F1) - d(P, F2)| = c,

where d(P, F1) is the distance between points P and F1, d(P, F2) is the distance between points P and F2andd(F1, F2) is the distance between F1 and F2.

F1 and F2 are the centers of the given circles and the foci of the hyperbola. Since only the tangent circles and their centers vary, and not the given circles or the distance between them, it's easy to see that the figure is always a hyperbola.