**TANGENT CIRCLES**

The most general form of the question: Construct
a circle tangent to two given circles.

**CASE ONE:**** A CIRCLE INSIDE ANOTHER CIRCLE IS GIVEN
**

__STEP SEVEN__: How we simplify figures is very important.
Let's hide what we don't need.

__STEP EIGHT__: Finally, animation part! I love it when
we animate things in GSP. Let's animate first
the key point X14 to convince ourselves that
the circle c4 is always tangent to c1 and
c2. I want to believe that! Therefore I will
click the figure below to see its animation:

**FINALLY, IT'S TIME TO TRACE THE LOCUS OF
THE CENTER OF THE TANGENT CIRCLE. **

__STEP NINE:__ The center of the tangent circle traces
out an ellipse with foci the centers of the
original circles. See figure below:

__STEP TEN__: I want to believe it is an ellipse: Click
the figure below:

**CASE TWO:**** TWO CIRCLES WITH NO COMMON POINTS ARE GIVEN**

We use exactly the same steps. We get an
animation file similar to the one described
in STEP EIGHT above. Click the figure below:

**IT'S TIME TO TRACE THE LOCUS OF THE CENTER
OF THE TANGENT CIRCLE.**

The center of the tangent circle traces out a hyperbola with foci the centers of the original circles. See figure below:

Now I want to believe it is a hyperbola: Click the figure below: