Tangent Circles

By Natalie Streiner

We are going to investigate the tangent circle to two
circles.

We will be looking at three cases specifically:

i.
When one given
circle lies completely inside the other

ii.
When the two
given circles over lap

iii.
When the two given
circles are disjoint

To help us with this investigation we will use the
script tool ÒCircle Tangent to Two CirclesÓ available in Assignment #5.

When one given circle lies completely inside anotherÉ

Here, the green circle is the circle tangent to the
two blue circles. We notice that this tangent circle lies completely inside the
larger circle and outside of the smaller circle, as well.

Now, letÕs see what happens when we trace the center
of our tangent circle.

We see that this trace forms an ellipse.

When two given circles overlapÉ

Once again, the green circle is our tangent circle.
Again we also see that the tangent circle lies inside of on of the circles it
is tangent to and outside the other.

Now, letÕs see what happens when we trace the center
of our tangent circle.

Just as the first case, this trace forms an ellipse,
this one being narrower than the first.

When the two given circles are disjointÉ

Here we see one of the disjoint circles lies
completely inside the tangent circle. When thinking about it, this makes sense.
If the circles are disjoint, and a tangent hits both of these circles at
exactly one, then it makes sense that one of the disjoint circles lays inside
the tangent circle. Now, letÕs see what happens when we trace the center of our
tangent circle.

Here, instead of an ellipse we see that the trace
forms a hyperbola. This hyperbola is formed since the circles are disjoint, and
the center of the tangent circle must trace two separate circles.