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
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny