What is a tangent circle? How is one constructed? These two questions
will be answered in the discussion below. For starters though, a tangent
circle is a circle that is tangent to another circle at one point. The simple
set of tangent circles occur when you only have two circles. This is shown
in **Figure 1** below.

The red circle is tangent to the blue circle in the above figure at point
**r**. This is a basic description of a tangent circle. In this discussion,
the tangent circles that will be discussed will be those that are tangent
to two other circles at the same time. The following is a step by step walk
through of how to construct a tangent circle that is tangent to two other
circles at the same time.

**1.**Given two circles and a point**R**on one of the two circles.**2.**Construct a line through the center of the circle(**c1)**that point**R**is on and point**R**.**3.**Construct a circle of with the same radius as the circle without point on it**R**using point**R**as its center and draw a segment between**c2**and the point(**P**) on the circle just constructed that is outside of the circle with point**R**on it and intersects the line that was drawn in step 2.**4.**Draw a segment between**c2**and**P**.**5.**Place a point at the midpoint of the segment just drawn and draw a perpendicular line to the segment just drawn through this point.**6.**Now place a point at the intersection the line drawn in step 5 and the one drawn in step 2. This point(**c3)**is the center of the tangent circle. Also, create a segment between point**c3**and**R**. This will be the radius of your tangent circle.**7.**Finally create a circle with center at**R**and with radius the length of the segment created in step 7.

With the construction of the tangent circle complete, now it is time
to look a little closer at the tangent circles themselves. As point **R**
moves around circle **c1**, what type of path would the center of the
tangent circle trace out? A circle? A hyperbola? An ellipse? **Figure 2**
illustrates the path of the center of circle **c3**.

As you can see from this figure, the loci of the centers of the tangent
circles(thick red line) trace out an ellipse. Why is it that the centers
of the tangent circles trace out an ellipse? The answer to lies in the construction
of the tangent circle. From the figure, where do the foci of the ellipse
look to be? One may guess that the foci are located at the centers of circles
**c1** and **c2**. This guess would be correct. To understand this,
look again at the figure associated with step 6 above. The distance from
**c1** to **R** is the radius of circle **c1**(in blue) and the
radius of **c2** is the same length as from points **R** to **P**
because that is how we defined it in step three. Now looking again at the
figure from step 6, the distance from point **c3** to **c2** is equal
to the distance from **c3** to **P**. This is true because point **c3**
was defined from a perpendicular line through the midpoint of the segment
between **c2** and **P**. Therefore, the distance from **c3** to
**c2** plus the distance from **c3** to **c1** is the same as the
sum the radius's of circles **c1** and **c2**. This is the defintion
for the foci of an ellipse.

What would you think the path of the loci of the centers of the tangent
circles would trace out if circles **c1** and **c2** intersected?
**Figure 3** shows this occurence.

Once again, an ellipse is formed. However, take a look at **Figure 4**.
Circles **c1** and **c2** still intersect but this time the loci of
the centers of the tangent circles forms a hyperbola. This occurs because
the tangent circle is tangent to the outside of circles **c1** and **c2**
at all points other than the area that the two circles intersect. In this
area the tangent circle is tangent to the inside of both circles.

There are many other investigations to do with tangent circles and if
you would like to explore some of your ideas go to **GSP**,
a tangent circle is already drawn and you can explore until your heart is
content.