This exploration stems from the following construction 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.
We begin with the construction of the desired circle:
We first draw circles A and C, with our chosen point E on A (It is not necessary that circle C be inside of A, but we must begin somewhere--we will discuss what happens when one circle is not inside the other later on).
Our goal is to find the center of a circle which will be tangent to both circles A and C. To be tangent to circle A at point E, the center of our circle must lie on the line EA. Having drawn this line, we draw a circle centered at E which is congruent to circle C. Let G be the intersection of EA and circle E.
Drawing circle E was the first part of the key to this construction. The second part comes from thinking about where our center must be in relation to points C and G. Since our circle must be tangent to both circle C and circle A, the distance from our center to circle C must be the same as the distance from our center to E. Then, by extending those segments a length equal to the radius of circle C, we see that our desired center must be equidistant from G and C. We find this point be constructing an isosceles triangle with segment GC as the base (We will do this be finding the midpoint of GC and constructing its perpendicular bisector. The desired point, I, will be where this bisector intersects the line AE).
Thus, we can draw circle I--it will go through E and J and will be tangent to our original circles:
One thing that can be explored is the locus of all such points I given all possible points E on circle A. You can open a GSP document which explores this by clicking here: centerlocus.gsp
Another interesting collection of points can be discovered be tracing the locus of the midpoint K of segment GI. The locus of these points is in Green below:
Notice the eggshape of this locus of points. To explore the locus of points created by this point K, for different sizes and positions of the original circles, click here: centerlocus2.gsp
Some of the more interesting loci occur when circle C is just outside of circle A. Also, note what happens when the circles intersect.