Arithmetic Mean Sangaku: What Is This About?
A Mathematical Droodle
What if applet does not run? 
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 © 19962017 Alexander Bogomolny
The applet is hopefully suggestive of the following sangaku [Bicycle, #39]:
Points C and D lie on a circle with diameter AB, 
What if applet does not run? 
We apply the Pythagorean theorem twice to the right triangles with hypotenuses joining the centers of the circles, as shown:
Let the distance from center O of the circumcircle to CD be d, the other centers I, P, Q, and the radii of the circles be R, r, r_{L}, r_{R}. Then the Pythagorean theorem gives
(r + d)^{2} + r^{2} = (R  r)^{2}, for r = r_{R} and (r  d)^{2} + r^{2} = (R  r)^{2}, for r = r_{L}, 
From which we express r_{R} and r_{L}:
(1) 
r_{R} = (R + d) + √2R(R + d) and r_{L} = (R  d) + √2R(R  d). 
which shows that the arithmetic mean of the two equals
(2)  (r_{R} + r_{L})/2 = R + (√2R(R + d) + √2R(R  d) )/2. 
One the other hand, we need to find the inradius r of the right ΔABC. For any right triangle with sides a, b and hypotenuse c, the inradius r can be found from
r = (a + b  c)/2. 
In our case, c = 2R, while a and b can again be found with the help of the Pythagorean theorem. It's not hard to see that one of them is √2R(R + d) whereas the other is √2R(R  d), which conforms with (2) and shows that the horizontal distance between P and Q is 2r. We may rewrite (1) as
(1') 
r_{R} = (R + d) + b r_{L} = (R  d) + a. 
Thus ST = 2r, where S and T are the points of tangency of (P) and (Q) with AB. In particular,
(3)  r = (r_{R} + r_{L})/2 = R + (a + b)/2. 
What remains to show is that the midpoint of ST is exactly the point of tangency W of the incircle (I) with AB, i.e.
Let's compute

And we are done.
(This sangaku has a nice generalization.)
References
 H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989
Write to:
Charles Babbage Research Center
P.O. Box 272, St. Norbert Postal Station
Winnipeg, MB
Canada R3V 1L6  J. Konhauser, D. Velleman, S. Wagon, Which Way Did the Bicycle Go?, MAA, 1996
Activities Contact Front page Contents Geometry
Copyright © 19962017 Alexander Bogomolny
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