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 ONE: Construct the given circles c1 and c2.
STEP TWO: Place movable points X1 and X2 on these circles. Then create lines d1 and d2 that cut the circles all the way through their diameters.
STEP THREE: Construct tha radia of the circles with legths r1 and r2.
STEP FOUR: Construct the circle c3 centered at X1 with radius r2. Construct the two intersection points of the circle c3 with the line d1. Remark: EX1=FX1=r2.
STEP FIVE: The aim is to find the center of that circle c4 which is tangent to both c1 and c2. From the figure above, we realize that that center must be somewhere on the line d1 that is as far from circle c1 as it is from circle c2. Therefore first we must construct the perpendicular bisector p of the line segment [BE]: The intersection point G of the bisector p with the line d1 is the center of the circle c4 with radius [GX1].
STEP SIX: How we name points is important. From the figure above, we see that X1 is the intersection point of c1with c4. Let's change its name to X14. Similarly let's give a name to the intersection point of the circle c2 with c4: X24.
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: