Ellipse Between Two Circles
The locus of centers of the circles inscribed in an arbelos is an ellipse. Arbelos  a crescent shaped figure  is formed by two tangent circles, one inside the other. Relaxing the tangency condition we obtain what may be called a blunt arbelos  an intermediate shape obtained by morphing an arbelos into an annulus.
A third circle can be inscribed into the shape touching the bigger circle internally and the smaller one externally. As with the arbelos, the centers of such circles lie on an ellipse.

What if applet does not run? 
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Copyright © 19962017 Alexander Bogomolny

What if applet does not run? 
Assuming the two circles have centers on the xaxis, one
Quite clearly the sum of distances of the center O of the inscribed circle to the centers of the two given circles is
The minor axis can be found by considering the extreme case of the inscribed circle with the center at the top point of the ellipse. This point along with the centers of
h² = ((r_{1} + r_{2}) / 2)²  ((x_{2}  x_{1}) / 2)². 
The minor axis of the ellipse is 2h.
Conic Sections > Ellipse
 What Is Ellipse?
 Analog device simulation for drawing ellipses
 Angle Bisectors in Ellipse
 Angle Bisectors in Ellipse II
 Between Major and Minor Circles
 Brianchon in Ellipse
 Butterflies in Ellipse
 Concyclic Points of Two Ellipses with Orthogonal Axes
 Conic in Hexagon
 Conjugate Diameters in Ellipse
 Dynamic construction of ellipse and other curves
 Ellipse Between Two Circles
 Ellipse in Arbelos
 Ellipse Touching Sides of Triangle at Midpoints
 Euclidean Construction of Center of Ellipse
 Euclidean Construction of Tangent to Ellipse
 Focal Definition of Ellipse
 Focus and Directrix of Ellipse
 From Foci to a Tangent in Ellipse
 Gergonne in Ellipse
 Pascal in Ellipse
 La Hire's Theorem in Ellipse
 Maximum Perimeter Property of the Incircle
 Optical Property of Ellipse
 Parallel Chords in Ellipse
 Poncelet Porism in Ellipses
 Reflections in Ellipse
 Three Squares and Two Ellipses
 Three Tangents, Three Chords in Ellipse
 Van Schooten's Locus Problem
 Two Circles, Ellipse, and Parallel Lines
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Copyright © 19962017 Alexander Bogomolny61185129 