In order to show that the path of the center of the tangent circle is a hyperbola, we need to show that the difference from the center of the tangent circle, I to two fixed points is a constant. Notice that points A and C are fixed since they are the centers of the two original circles. Notice also that the radii of the large circle and the small circle is AE and CM respectively. We can see from the picture above that IE = IM since they are both radii of the tangent circle. In addition, IE=EA+AI and IM=IL+LM by segment addition. We can also observe that IC-CM=IM. This implies, by substitution, that IC-CM=IA+AE. Thus, by algebra, we can get that IC-IA=AE+CM. Since the radii, AE and CM are constants, their sum is also a constant. Therefore we can see that the difference of distance from the center of the tangent circle, I to the center of the small circle, C and the distance from I to the center of the larger circle A is always equal to a constant. We have now shown that the path of the center of the tangent circle is a hyperbola if the small circle is outside of the large circle.
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