From the Calculus, we know that an ellipse is the set of all points in the plane such that the sum of whose distances from two fixed points is a constant. In this applet, we are going to show that the locus of the center of a circle that is tangent to two given circles is an ellipse.
Let the two following circles be given:
C1:
C2:
Then plot a point C on the circle C2. By intuition, we know that there exists a circle that is tangent to both C1 and C2, and passes through the point C. We can parameterize the point C by the angle α that the ray OC forms with the positive x-axis. Then we can represent the point C by (3+(1/2)cos(α),(1/2)sin(α))
By clicking 'Show the tangent circle', you can display this tangent circle.
Now, by clicking ‘Start animation’, you can see how the tangent circle looks like as α changes.
By clicking ‘Show the relationship between points’, you can find the property of this tangent circle.
Then by clicking ‘Show the trace of the center’, you can visualize the locus of the center of tangent circle, which looks like an ellipse.
Finally, by clicking ‘Show the locus’, you can understand why the locus is an ellipse.

By clicking 'Clear traces', you can clear the trace of the center of the tangent circle.
By clicking 'Reset', you can reset.