Ellipse in Arbelos
Here is a problem (E 762) from the 1947 American Mathematical Monthly. It was proposed by J. R. Van Andel, Naval Air Experimental Station, Philadelphia, Pa. (The solution below is by Norman Anning, Ann Arbor, Michigan):
Let A_{1} and A_{2} be two circles with radii a_{1} and a_{2} and centers


What if applet does not run? 
References
 J. R. Van Andel,; Norman Anning, American Mathematical Monthly, Vol. 54, No. 9. (Nov., 1947), pp. 547548.
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander BogomolnyPart (a) is elementary. A glance at a figure shows that the center of C is always in such a position that the sum of its distances from
That the major semiaxis of the ellipse is
h² = [(a_{1} + a_{2})/2]²  [(a_{2}  a_{1})/2]², 
so that indeed, h² = a_{1}a_{2}. (These two facts are easily extended to a more general shape.)
What if applet does not run? 
For part (b), let (X, Y) be a generic point on C^{t}. Then
(1)  (X² + Y²)·φ_{t}  2a_{1}a_{2}(a_{1} + a_{2})X  4t a_{1}a_{2}(a_{2}  a_{1})Y + 4(a_{1}a_{2})² = 0. 
Apply to C_{t} the inversion
X = 4a_{1}a_{2} x / (x² + y²), Y = 4a_{1}a_{2} y / (x² + y²). 
Then (1) becomes
x² + y²  2(a_{1} + a_{2})x  4t(a_{2}  a_{1})y + 4φ_{t} = 0, 
which may be rewritten as
(2)  (x  a_{1}  a_{2})² + (y  2ta_{2} + 2ta_{1})² = (a_{2}  a_{1})². 
With t as parameter, (2) is the family of equal circles which touch the parallel lines
Now invert again. Circle C_{0} inverts into itself and (2) inverts into (1). The line
The solution in the Monthly is followed by the following note:
One tracing the history of the problem would find it under arbelos. See R. Johnson's Modern Geometry, for instance. The neatest of the properties,
The proposer pointed out that J. Steiner in Geometrische Betrachtungen (1826) discussed, in particular, the chains of circles corresponding to the sequences
 Arbelos  the Shoemaker's Knife
 7 = 2 + 5 Sangaku
 Another Pair of Twins in Arbelos
 Archimedes' Quadruplets
 Archimedes' Twin Circles and a Brother
 Book of Lemmas: Proposition 5
 Book of Lemmas: Proposition 6
 Chain of Inscribed Circles
 Concurrency in Arbelos
 Concyclic Points in Arbelos
 Ellipse in Arbelos
 Gothic Arc
 Pappus Sangaku
 Rectangle in Arbelos
 Squares in Arbelos
 The Area of Arbelos
 Twin Segments in Arbelos
 Two Arbelos, Two Chains
 A Newly Born Pair of Siblings to Archimedes' Twins
 Concurrence in Arbelos
 Arbelos' Morsels
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
Conic Sections > Ellipse
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny68545852