This exploration looks at creating tangent circles. A script tool for creating tangent circles can be accessed here. We are interseted in following the centers of each of the two tangent circles as their point of intersection traverses one of the original circles. There are three cases to consider:
One of the original circles is contained within the other. In this case the locus of points created by the two tangent circles are concentric ellipses. In the diagram below the two green circles are our originals, the red and blue circles the tangent circles, and the thick lined red and blue curves the loci of the tangent circles' centers ( respective to their colors).
When the two original circles intersect one another we see below that the loci of centers is given by one ellipse and one hyperbola.
In the case that the two original circles are non intersecting and one is not contained within the other the resulting loci of tangent circle centers is two hyperbolas.
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