 Tangent Circles

I will investigate a circle tangent to two given circles.

First, let’s start with what we are given: two circles, c1 and c2, one inside the other with centers A and B. Next, I will place a sliding point on each circle. Let’s label those points C and D. Now, I will construct a line segment from B to D, which is the radius of radius c2. I will also create a line going through points A and D. Next, I will construct a circle with a center at the sliding point C, with radius BD. Let’s name this circle c3. Now, I will construct the intersection of circle c3 and line AD, on the exterior of circle c1. Let’s label this intersection point, E. Now, I will draw a line segment BE and construct its midpoint M. Draw a perpendicular line through BE at M. The intersection of this perpendicular line and line AD, will be named F. Now, I can construct another circle, c4, with center at F and radius FC. This new circle is the circle that is tangent to the two given circles. Now, if I hide all of the additional constructions, I can see the tangent circles more clearly. What will happen if I slide point C around the circle? If I animate point C, then some intersecting things happen.

First, the circle always remain tangent, but the radii of the circle centered at F changes as C moves around the circle. Also, let’s trace F as D is animated. I see that point F moves in an ellipse with foci A and B. 