Square and Circle in a Gothic Cupola
This is another in a series of simple sangaku problems that is solved by a repeated application of the Pythagorean theorem.
Two quarter circles inscribed in a square form a gothic cupola. Inscribed in the latter is a circle on top which stands a small circle. What is the relation between the radius of the circle and the side of the small square and the side the big square?
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Copyright © 19962018 Alexander Bogomolny
This is another in a series of simple sangaku problems that is solved by a repeated application of the Pythagorean theorem.
Two quarter circles inscribed in a square form a gothic cupola. Inscribed in the latter is a circle on top which stands a small circle. What is the relation between the radius of the circle and the side of the small square and the side the big square?
Solution
With the reference to the diagram, assume the side of the big square is 1, x the side of the small square and r is the radius of the circle in question.
In right ΔAFG, AG² = AF² + FG²:
1 = (1/2 + x/2)_{2} + x_{2},
which simplifies to a quadratic equation
5x² + 2x  3 = 0,
with a single positive root, x = 3/5. The Pythagorean theorem applied to ΔAMO,
(1  r)² = (1/2)² + (x + r)²,
which, with x = 3/5, simplifies to a linear equation in r so that
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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Contact Front page Contents Geometry Up
Copyright © 19962018 Alexander Bogomolny