Given two circles and a point I will construct a circle tangent to the two circles with one point of tangency being the designated point. View the GSP construction for Tangent Circle.
The construction was made by selecting a point on the outer circle and making a circle equal to the smaller circle around that point. Then I connected the center of the inner circle to the point on the similar circle that lies on the diameter of the large circle. Then I found the perpendicular bisector of this segment and the point where the bisector intersects the diameter of the larger circle. This intersection points now becomes the center of the tangent circle and radius of this circle is the center to the point on the larger circle.
Another case is when the tangent circle engulfs the inner circle. View the GSP construction here.
In this case we find the perpendicular bisector of the segment with endpoints center of inner circle and the point, on the similar circle that was constructed, that lies on the diameter of the larger circle on the inside of the larger circle. This gives us a tangent circle that fully engulfs the inner circle.
Now let us look at the loci of the centers of the tangent circles for both of the above mentioned cases.
This allows us to see that the loci of the center of the tangent circle is an ellipse with foci at the centers of the two circles, the inner and the outer circle.
We can see from this construction that the loci of the center of the tangent circle in this case is also and ellipse. This time the ellipse is within the inner circle and its foci are still the centers of the inner and outer circles.
Now lets look at the tangent circles when the two given circles intersect.
Looking at the path of the tangent circle we can see that once tangent circle reaches the point where the circle is outside the outer circle the tangent circle changes its path to being tangent on the inside of the inner circle and outside of the larger circle simultaneously.
The loci of the tangent circle is this case is still an ellipse with foci as the two centers of the given circles.
In case 2 the path of tangent circle is the exact opposite as in case 1. The tangent circle changes paths in the part where the smaller circle is inside the larger circle. Interestingly, the loci is no longer an ellipse in this case. It is now parabolic keeping the foci as the centers of the two given circles.
What if the two given circles are disjoint?
When the two given circles are disjoint the tangent circles is tangent to both circles simultaneously on the outside. The loci of the center of the tangent circle in this case is also parabolic with foci the center of the two given circles.
Again in this case the tangent circle is tangent to both given circles in the area outside both circles. The loci is a more stretched out parabola with foci staying as the centers of the two given circles.
Tangent Line traces:
The tangent line traces coincide with the trace of the loci of the tangent circles for each case.
Lets see how the locus of the midpoint of the segment that formed the base of the key isosceles triangle behaves.
When I traced the midpoint of the base of the isosceles I noticed that it creates a circle and no matter where we place the smaller circle, the trace circle remains the same size in both cases.