Five Incircles in a Square: What Is This About?
A Mathematical Droodle
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Copyright © 19962018 Alexander Bogomolny
Five Incircles in a Square
Here we have a very characteristic sangaku  a Temple geometry  problem. Most of geometric sangaku dealt with several circles inscribed in other circles or other shapes. In addition, most often a sangaku was a computational problem as opposed to problems that require a proof. The problem below is a little of both: a proof of the result is supported by a chain of calculations.
In the manner of proof #3 of the Pythagorean theorem, four equal right triangles and a small square are combined into a larger square. Circles are inscribed into the four triangles and the inner square. A question: What can be said of the configuration where all five circles have equal radii?
So assume the triangles have legs a and b
r = (a + b  c)/2.
On the other hand, the radius of the circle inscribed into the inner square is obviously
r = (a  b)/2.
Equating the two we get
c = 2b,
which means that the angle opposite b is 30° and the other one is 60°. From here we easily find the common inradius:
Most of the sangaku problems are much more difficult.
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
Sangaku

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