By Jaepil Han
1. Make GSP constructions and script tools for construction of the tangent circles. An appropriately constructed GSP Script Tool will enable you to investigate all the cases:
Here's the script tools for the tangent circles and its centers.
Here's my library of script tools in Assignment 5.
(i) When one given circle lies completely inside the other
CT1 and CT2 represent the centers of the tangent circles.
The locus of the center, CT1, of the tangent circle looks like an ellipse.
Meanwhile, the locus of the center, CT2, of the tangent circle looks like an narrower ellipse than the other one.
Here's an interesting thing. When the center of the inside circle getting close to the center of the outer circle, the loci of the two centers, CT1 and CT2, are getting circular shapes. This conjecture still need to be proved, so it might be a good for further investigation.
(ii) When the two given circles overlap
Similarly, the locus of the center, CT1, of the tangent circle looks like an ellipse, which is passing through the two intersections of the given circles.
Also, the locus of the center, CT2, of the other tangent circle looks like two parabolas or a hyperbola. To investigate more, I've placed two circles differently to make the overlapping area varied.
When the circles placed overlapping one of the circles shares its area mostly with the other, the locus of the center, CT2, of the other tangent circle looks like an angle-bracket-shaped hyperbola.
Also, when the two given circles overlaps their areas slightly, the locus of the center, CT2, is almost two parallel lines.
(Technically, those are not parallel lines, but I couldn't find any other good expression that describes this circumstance.)
(iii) When the two given circles are disjoint
Here's the two tangent circles when the two given circles are disjoint, completely not sharing any part of them.
The locus of the center, CT1, shapes a somewhat typical hyperbola. At this point, we cannot determine whether it is really a hyperbola or not.
Meanwhile, the locus of the other center, CT2, shapes sort of two lines like the case of given circles only share a small area of them.
Consequently, the loci of the centers of the tangent circles might shape a circle, an ellipse or a hyperbola, depending on the relation between the two given circles.
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