Arithmetic Mean Sangaku: What Is This About?
A Mathematical Droodle
What if applet does not run? 
Sangaku

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Activities Contact Front page Contents Geometry
Copyright © 19962018 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 © 19962018 Alexander Bogomolny