Assignment
7
TANGENT
LINES

Gives
these two circles, I will create a circle that is tangent to both
of them.

I
will first construct a radius in the smaller circle.

Next
I will construct a line through the center of the larger circle.

Now
I will copy the smaller circle so that the center the point on
the top of the larger circle .

Now
I will connect the top point of the new circle with the center
of the smaller circle.
The
result is the orange segment.

I
will now create the perpendicular bisector of the orange segment.
The intersection of the perpendicular bisector and the dashed
blue line will be the center of the tangent circle.
The
result is this dotted red circle.

Through
the construction of several perpendiculars, I have created the
inside dotted circle, which is tangent to both initial circles.

Click
here to see what happens
when the circles are rotated around each other and the point of
intersection is traced.

Here
is the still picture of what you just saw.
So,
as you can see, the traced center of the tangent circle creates
an ellipse. This seems to make sense since points A and B would
be considered foci. This follows the definition of an ellipse.

What
happens if the two circles initially intersect eachother?

After
constructing the same process as above, check out the animation
for this example by clicking HERE.

Here
is the still picture of the locus of points.
The
red ellipse is the traaced object. Again we can see that the foci
must be the two points near the ellipse in the interior.

The last
situation is where the circle is outside the initial circle.

I
will construct the circle tangent to both circles in the same
fashion that I did above.
Notice that the red dotted circle is tangent to both green circles.

Click
HERE to see the locus
when the intersection is traced around the circle.

Here
is the still picture of what you just saw.
In
this situation we get a hyperbola instead of an ellipse. So here
we can see that the foci must be the centers of the two green
circles.

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