*CASE 1*

*Let's start the following
problem : Construct a circle tangent to the given two circles.*

*First of all, draw a line
passing through the center of the one circle. Let's call B' the
point of the intersection between the circle and the line. Then,
measure the radius of the other circle and draw a circle with
the same radius at the point B'.*

*Next, take the point c,
which is the point of intersection between the line AB' and the
new circle. Construct the segment BC.*

*After that, find the midpoint
D between the segment BC.*

*Now, draw a pependicular
line at D on the segment BC.*

*Can you find the point E?
It is the point of intersection between two lines. The point E
is the center of the circle tangent to the given two circle. Take
the radius from E to B' and consturct a tangent circle.*

*Here is the GSP file. Click* here and explore the tangent circles. Get the animation
to see the trace of the center of the tangent circles.

*Do you see an ellipse?*

*CASE 2*

*Let's think about the other
cases?*

*If two given circles intersect,
we can construct the tangent circle as below. Guess what the trace
of the center of the tangent circles is? Yes, we have an ellipse
again in this case.*

Please click here to get the GSP file.

*However, can we cover all
tangent circles? No. We need to investigate the case below. Here
the trace is a hyperbola.*

*CASE 3*

*Well, now let's look at
the tangent circles when two given circles are disjoint. We can
generate two animaitons. In the first one, the trace of the center
of the tangent circles is the hyperbola while as we have the ellipse
in the second animaiton.*