Equal Incircles Theorem
What Is It About?
A Mathematical Droodle

You probably do not see the applet because nowadays no browser supports Java. Hopefully a solution could eventually be found. Meanwhile, here's what you would see if the applet was working:

Equal incircles theorem, the case of three layers

(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.)

Explanation

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 M₁AM₂, M₂AM₃, ..., M_N-1AM_N all have equal radii. Then the same is true of triangles M₁AM₃, M₂AM₄, ..., M_N-2AM_N, and also of triangles M₁AM₄, M₂AM₅, ..., M_N-3AM_N, and so on.

Equal incircles theorem, the case of four layers

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 2ⁿ 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

AM₂ = √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_iAM_i+1 is equal to that of ΔM_i+2AM_i+3, i = 1, 2, ..., n-3. Then the incircles of the "second level" triangles M_iAM_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

71931240

Equal Incircles TheoremWhat Is It About? A Mathematical Droodle

You probably do not see the applet because nowadays no browser supports Java. Hopefully a solution could eventually be found. Meanwhile, here's what you would see if the applet was working:

Equal Incircles Theorem

References

Sangaku

Equal Incircles Theorem
What Is It About?
A Mathematical Droodle