Tangent Circles

By Taylor Adams

Given two circles, C_{1 }and
C_{2}, let’s find a circle tangent to both.

First, construct a circle with its
center on C_{2} with the same radius as C_{1}.

Next, construct a line through the
center of C_{2} and the center of the new circle, point C. The intersection of this line with the new
circle outside of circle C_{2} is labeled point D> This is can be thought of as a line
through the diameter of C_{2}.

Next, construct a line segment from the
center of C_{1}, point A, and point D.

Next, construct the perpendicular
bisector of line segment AD. Label the
intersection of the perpendicular bisector with the diameter of C_{2}
as point E. This point will be the
center of the circle tangent to both circles C_{1} and C_{2}.

Finally, construct a circle with
center E with radius EC. This is the
circle that is tangent to both circles C_{1} and C_{2}.

Let’s look at the loci of the centers
of the tangent circle.

To do this, we can animate the
circles in GSP and trace the tangent circle’s center to form the loci of the
centers.

When one circle is inside of the
other, we can see that the loci of the center is an oval shape.

We want to prove that the loci is an
ellipse. To do this, we want to show
that the distance from a fixed point to the center of the tangent circle to
another fixed point is constant. We will
use the centers of the two original circles as the fixed points.

This gsp file shows that this distance is constant.

When the two circles cross each
other, we can see that the loci of the center of the tangent circle is an oval
shape.

We want to show that it actually is
an ellipse. To do this, we want to show
that the distance from a fixed point to the center of the tangent circle to
another fixed point is constant. We will
use the centers of the two original circles as the fixed points.

This gsp file shows that this distance is constant.

When the two circles are disjoint,
the loci of the center of the tangent circle creates a hyperbola.

To prove that this acutally is a hyperbola, we want to show that the
difference of the distance from a fixed point to the center of the tangent
circle and the distance from a second fixed point to the center of the tangent
circle is constant. We will use the
centers of the two original circles as the fixed points.

This gsp file shows that the difference of these distances is constant.