Four Touching Circles II
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A Mathematical Droodle
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Copyright © 19962018 Alexander Bogomolny
Four Touching Circles II
The applet suggests the following theorem:
(A)  Let there be 4 circles S_{1}, S_{2}, S_{3}, and S_{4} each tangent cyclically to its neighbors, so that S_{1} touches S_{2} and S_{4}, S_{2} also touches S_{3}, and the latter touches S_{4}. Prove that the four tangency points are concyclic, i.e. lie on a circle. 
It has been observed that the theorem only holds when the number of external tangenices of the four circles is even. If the number of external tangenices is odd, the theorem no longer holds.
The theorem has been proved elsewhere.
The sides of the quadrilateral in question are the chords of the circles S_{1}, S_{2}, S_{3}, and S_{4}. A is the point of tangency of S_{1}, S_{2}, B that of S_{2}, S_{3}, and similar for C and D. The angles of the quadrilateral at A, B, C, D are α_{1}, α_{2} and so on.
In a couple of possible configurations just testing the angles of ABCD shows that the quadrilateral is cyclic.

}→  α_{1} + α_{3} = α_{2} + α_{4}  
 }→  α_{1} + α_{3} = α_{2} + α_{4}  
 }→ ? 
where β_{i} is the angle formed by the chord in S_{i} and the common tangent with either S_{i1} or S_{i+1} (the indices taken modulo 4).
In the first two cases it is obvious that
In the third case, the strategy should be changed as, otherwise, we come up with a useless identity
In the third case, we proceed as follows:
α_{1} + α_{4}  = (β_{1} + β_{4}) + (β_{4} + β_{3})  
= β_{1} + β_{3} + 2β_{4},  
α_{2} + α_{3}  = (β_{2} + β_{1}) + (β_{3} + β_{2})  
= β_{1} + 2β_{2}  β_{3}. 
Now, since for the selfintersecting quadrilateral in the last diagram
Relevant pages:
 The Four Touching Circles Problem
 The Four Touching Circles Problem II
 The Four Touching Circles Problem III
 The Four Touching Circles Problem IV
Inversion  Introduction
 Angle Preservation Property
 Apollonian Circles Theorem
 Archimedes' Twin Circles and a Brother
 Bisectal Circle
 Chain of Inscribed Circles
 Circle Inscribed in a Circular Segment
 Circle Inversion: Reflection in a Circle
 Circle Inversion Tool
 Feuerbach's Theorem: a Proof
 Four Touching Circles
 Hart's Inversor
 Inversion in the Incircle
 Inversion with a Negative Power
 Miquel's Theorem for Circles
 Peaucellier Linkage
 Polar Circle
 Poles and Polars
 Ptolemy by Inversion
 Radical Axis of Circles Inscribed in a Circular Segment
 Steiner's porism
 Stereographic Projection and Inversion
 Tangent Circles and an Isosceles Triangle
 Tangent Circles and an Isosceles Triangle II
 Three Tangents, Three Secants
 Viviani by Inversion
 Simultaneous Diameters in Concurrent Circles
 An Euclidean Construction with Inversion
 Construction and Properties of Mixtilinear Incircles
 Two Quadruplets of Concyclic Points
 Seven and the Eighth Circle Theorem
 Invert Two Circles Into Equal Ones
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
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