In the interactive model below, press 'Animate Point C.' Notice that as point C moves, the red circle remains tangent to the blue circles. To draw another curve, press the button again to stop the tracing. Drag point A or point B to change the size and placement of the two circles. Then clear the existing trace by clicking the red 'X' in the lower right corner. Animate point C again.
Can you prove that point C traces an ellipse using the distance definition of an ellipse?
For a closely related construction, see Tangent Circles, Part 2.