## The Loci of the Centers of the Tangent Circles

#### by: BJ Jackson

This discussion is based on the following picture with A the
center of the red tangent circle, B the center of the blue tangent
circle and C the arbitrary point used to construct the circles.

Now, we want to figure out what happens to A and B when C moves
around the circle. Using GSP, there are two differnt ways to look
at this. First, one can use the animation feature and animate
C around the circle while tracing the points A and B. This is
the method That I am going to use. The second is to use the locus
feature of GSP. Either way, one will get the same picture. I choose
the animate feature because it allows one to see the locus being
constucted. The locus command just creates the locus without the
user seeing where it came from. So, based on other constuctructions,
one would expect the loci to be an ellipse. Which as can be seen
below, the loci are ellipses.

What happens if the circles intersect? Would one expect the
loci to remain ellipses? Here, the locus of A does remain an ellipse
but the locus of B changes into an hyperbola which can be seen
below.

Now, what happens if the original circles are disjoint? Will
we get two ellipses? An ellipse and a hyperbola? Or something
else? In this case, one should get two hyperbolas. A picture of
this is below.

One other loci that I found interesting are the midpoints of
the segments used to make the perpendiculars that found the center
of the tangent circles. These points are now marked D and E. I
expected their loci to work the same fashion as the points A and
B. To start with two ellipses, then get an ellipse and an hyperbola,
and finally to have two hyperbolas. When I found these loce, it
was surprising to find out they are circles in all three of the
different situtations. An example of this is below.

Click **here **to link
to a GSP sketch that will allow one to manipulate the circle and
see all of the different loci.

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