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 (17311807) 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, r_{a}, r_{b}, r_{c} be the radii of the circumcircle (O), the incircle (I) and the excircles (I_{a}), (I_{b}) and (I_{c}). The fourth tangents common to pairs of the excircles (I_{a}), (I_{b}) and (I_{c}) form a triangle A'B'C'. Show that centers of the incircle of ΔA'B'C' and the circumcircle of ΔI_{a}I_{b}I_{c} coincide and that the radius r' of the former satisfies:

The result 2r' = r + r_{a} + r_{b} + r_{c} 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.
What if applet does not run? 
References
 Fukagawa Hidetoshi, Tony Rothman, Sacred Mathematics  Japanese Temple Geometry, Princeton University Press, 2008, pp. 298299
Sangaku
 Sangaku: Reflections on the Phenomenon
 Critique of My View and a Response
 1 + 27 = 12 + 16 Sangaku
 345 Triangle by a Kid
 7 = 2 + 5 Sangaku
 A 49^{th} Degree Challenge
 A Geometric Mean Sangaku
 A Hard but Important Sangaku
 A Restored Sangaku Problem
 A Sangaku: Two Unrelated Circles
 A Sangaku by a Teen
 A Sangaku FollowUp on an Archimedes' Lemma
 A Sangaku with an Egyptian Attachment
 A Sangaku with Many Circles and Some
 A Sushi Morsel
 An Old Japanese Theorem
 Archimedes Twins in the Edo Period
 Arithmetic Mean Sangaku
 Bottema Shatters Japan's Seclusion
 Chain of Circles on a Chord
 Circles and Semicircles in Rectangle
 Circles in a Circular Segment
 Circles Lined on the Legs of a Right Triangle
 Equal Incircles Theorem
 Equilateral Triangle, Straight Line and Tangent Circles
 Equilateral Triangles and Incircles in a Square
 Five Incircles in a Square
 Four Hinged Squares
 Four Incircles in Equilateral Triangle
 Gion Shrine Problem
 Harmonic Mean Sangaku
 Heron's Problem
 In the Wasan Spirit
 Incenters in Cyclic Quadrilateral
 Japanese Art and Mathematics
 Malfatti's Problem
 Maximal Properties of the Pythagorean Relation
 Neuberg Sangaku
 Out of Pentagon Sangaku
 Peacock Tail Sangaku
 Pentagon Proportions Sangaku
 Proportions in Square
 Pythagoras and Vecten Break Japan's Isolation
 Radius of a Circle by Paper Folding
 Review of Sacred Mathematics
 Sangaku à la V. Thebault
 Sangaku and The Egyptian Triangle
 Sangaku in a Square
 Sangaku Iterations, Is it Wasan?
 Sangaku with 8 Circles
 Sangaku with Angle between a Tangent and a Chord
 Sangaku with Quadratic Optimization
 Sangaku with Three Mixtilinear Circles
 Sangaku with Versines
 Sangakus with a Mixtilinear Circle
 Sequences of Touching Circles
 Square and Circle in a Gothic Cupola
 Steiner's Sangaku
 Tangent Circles and an Isosceles Triangle
 The Squinting Eyes Theorem
 Three Incircles In a Right Triangle
 Three Squares and Two Ellipses
 Three Tangent Circles Sangaku
 Triangles, Squares and Areas from Temple Geometry
 Two Arbelos, Two Chains
 Two Circles in an Angle
 Two Sangaku with Equal Incircles
 Another Sangaku in Square
 Sangaku via Peru
 FJG Capitan's Sangaku
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
68007246