We begin this investigation with the following problem.
Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point.
After some analysis ( and some help) we constructed a GSP Script tool which will be used to further investigate the problem.
The red circle was constructed tangent to the two given green circles
If we look at the locus of the center we note that it is an ellipse. The GSP File will give you a better picture.
This is another case of tangent circles. A script tool is provided.
The locus of the center of this tangent circle is also an ellipse. A GSP file is provided for you to investigate.
Using the script tools to further investigate, we note that the previous constructions holds when the given circles intersect. However, if the two given circles intersect, we have two cases for the locus of the center of the tangent circle:
The first case is an ellipse whereas the second case we have hyperboles.
If the circles are disjoint we see a number of cases all the cases seemed to be hyperbolas but there there is also what appears to be parallel lines. Although this is inconclusive.
Here is another case of the locus of centerPlease look at the GSP files and continue this investigation.