## Investigations with
Tangent Circles

### something I discovered

### by: Lauren Wright

In this investigation, we take a look
at some properties of tangent circles. In order to explore these
properties, we will need to work with two main constructions.
Given two circles and a point, P, on one of the circles, construct
a circle tangent to the two circles with one point of tangency
being P.
One possible construction would make
the smaller circle external to the tangent circle.
Another possible construction would
make the smaller circle internal to the tangent circle.
These are the two possible tangent
circles for this construction. To see the motivation behind these
constructions, click
here to go to Dr. Jim Wilson's Assignment 7 Page.
OR, click
here to download a file with the two constructions and a script
tool in it.

Now, let's take a look at the loci
of the centers of the tangent circles.
The loci of the centers of the tangent
circles take the shape of an ellipse. An ellipse is the loci of
all points whose sum of the distances from two foci is a constant.
In this case, the two foci would be the centers of the original circles.
What does THAT mean? It means that
no matter where the center of the tangent circle is, the sum of
its distances from the two foci will remain constant. Click
here to explore this relationship by moving the point of tangency,
P, around the original circle.

Now, the obvious question is, will
the loci of the centers of the tangent circles always form an
ellipse? Let's take a look.
CASE 1: What happens if the two original
circles intersect?
This time, the loci of the blue circle
is an ellipse. BUT, the loci of the red circle is a hyperbola.
The foci are still the centers of the two original circles.
What is the relationship of the foci
with a hyperbola? Well, a hyperbola is the loci of all points
whose difference of the distances from the two loci is a constant.
Click here to explore
this relationship by moving the point of tangency, P, around the
original circle.

CASE 2: What happens if the two original
circles are disjoint?
Here we see that the loci of the centers
of the tangent circles are both hyperbolas. Click
here to explore this relationship.

After exploring the paths of these
loci, I found that they had formed two of the conic sections -
the ellipse and the hyperbola. So, my next question was, "Can
I change the original circles so that the loci form the other
significant conic section, the parabola?"
To do this, I manipulated the sketch
above until the loci formed a parabola on the screen. When the
parabolas appeared on the screen, the two original circles were
disjoint and I had made one of them really large.
In this picture, you cannot even see
the center of the circle that I made large. The large circle appears
to be approaching a line as I drag the center further and further.
Recall that the foci of the two hyperbolas
are the centers of the two original circles. My conclusion is
that as one foci approaches infinity, the loci will approach a
parabola. In other words, the loci forming a parabola is a limiting
case.

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