Neuberg Sangaku

Many of sangaku problems have been thought up in the West, sometimes earlier and sometimes later than they have been published in Japan. For researchers into the Japanese mathematics (wasan) of the Edo period this is of course important to identify the instances where precedence lies in the East. A construction of three tangent circles in what's now known as Malfatti's problem after an 1803 article by Gian Francesco Malfatti (1731-1807) has been proposed some 30 years ealier by the great Japanese mathematican Ajima Naonobu.

The applet below presents a problem posed by J. J. B. Neurberg in 1896 (Methesis, p. 193, problem 1078):

 

Given a ΔABC, let R, r, ra, rb, rc be the radii of the circumcircle (O), the incircle (I) and the excircles (Ia), (Ib) and (Ic). The fourth tangents common to pairs of the excircles (Ia), (Ib) and (Ic) form a triangle A'B'C'. Show that centers of the incircle of ΔA'B'C' and the circumcircle of ΔIaIbIc coincide and that the radius r' of the former satisfies:

  r' = 2R + r = (r + ra + rb + rc)/2.

The result 2r' = r + ra + rb + rc is known to have been presented in 1803 by Yamamoto Norihisa on a sangaku at the Echigo Hukasan shrine, Niigata prefecture. The tablet has been lost but was recorded in Nakamura's 1830 manuscript Saishi Shinzan.

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


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References

  1. Fukagawa Hidetoshi, Tony Rothman, Sacred Mathematics - Japanese Temple Geometry, Princeton University Press, 2008, pp. 298-299

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