Equal Incircles Theorem
What Is It About?
A Mathematical Droodle
What if applet does not run? 
(The number of points on the base line which is originally 4 can be changed by clicking on it a little off its center line.)
Activities Contact Front page Contents Geometry Eye opener
Copyright © 19962018 Alexander Bogomolny
Equal Incircles Theorem
The applet suggests the following theorem [Wells, p. 67] from triangle geometry:
Let A be a point. Assume points M_{i}, i = 1, 2, ..., N (N > 3) lie on a line not through A. Assume further that the incircles of triangles M_{1}AM_{2}, M_{2}AM_{3}, ..., M_{N1}AM_{N} all have equal radii. Then the same is true of triangles M_{1}AM_{3}, M_{2}AM_{4}, ..., M_{N2}AM_{N}, and also of triangles M_{1}AM_{4}, M_{2}AM_{5}, ..., M_{N3}AM_{N}, and so on.
I do not know of an elegant proof of that theorem. In [Wells, p. 67] it appears without a proof. There's no reference either.
The problem has been also discussed by R. Honsberger [Delights, section 17], where he mistakenly claims that the equality of the incircles only extends on the "triangles formed by sets of 2^{n} consecutive triangles in the fan". The solution, nonetheless, works for a more general statement. Honsberger found this to be #2.2.5 in the Sangaku collection by H. Fukagawa and D. Pedoe. In fact problem #2.2.5 shows only two circles and the question is to find the length of the common tangent from the apex A in case the two circles are equal. In this form the problem appears on a surviving 1897 tablet from the Chiba prefecture. The answer to this problem can be found to be
AM_{2} = √s(s  a),
where a is the base and s is the semiperimeter of the triangle.
However, the theorem is a consequence (and also a generalization) of a theorem published in 1986 by H. Demir, which, as was shown by J. B. Tabov, admits if not a more elegant proof, then quite an elegant generalization:
Assume n > 4, and the inradii of the "first level" triangles are equal in pairs: the inradius of ΔM_{i}AM_{i+1} is equal to that of ΔM_{i+2}AM_{i+3}, i = 1, 2, ..., n3. Then the incircles of the "second level" triangles M_{i}AM_{i+2} are all equal.
(There is another, simple proof of the general statement.)
References
 H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989
 R. Honsberger, Mathematical Delights, MAA, 2004
 D. Wells, Curious and Interesting Geometry, Penguin Books, 1991
Sangaku

[an error occurred while processing this directive]
Activities Contact Front page Contents Geometry Eye opener
Copyright © 19962018 Alexander Bogomolny