Let us consider the problem of having two circles whereas one is inscribed within the other, and a third circle is constructed such that it is tangent to both the original circles.
We are given :
A point on one of the circles
Let us start with two circles below shown in green.
We add the red circle inside the large green circle keeping it outside of the smaller green circle. The red circle has been constructed using the point on the outer circle.
When adding the purple circle tangent to the larger green circle we observe that there is no tangency to the smaller circle. The purple circle is constructed to circumscribe the smaller green circle.
If we look at the loci of the centers of the purple tangent circle, what shape would be graphed? To see click HERE!
Each set of loci makes a separate ellipse.
Loci of the base of the isosceles triangle
Review the construction of the red tangent circle, the one that was externally tangent to the small green circle (above).
Segment AB is the base of an isosceles triangle. It is apparent that all points on the dotted red line are equidistant from both point A and B since the line is the perpendicular bisector of AB.
Can you guess what shape would be made with the trace of the loci of the midpoint of the base of the isosceles triangle? Instead of making an ellipse, the loci of the midpoints is a ________?
(Try this and see)
To look at the locus of the midpoint of the segment, click HERE!
Notice that the locus of the midpoints of the purple circle is also the same.
The consistency of the arrangement of the points is quite interesting.
Conclusion: In summary, the loci of the centers of the tangent circles form ellipses. The loci of the midpoints of the base of the isosceles triangle (while under construction) form circles.