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?

Solution 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 + x2,

which simplifies to a quadratic equation

5x² + 2x - 3 = 0,

with a single positive root, x = 3/5. The Pythagorean theorem applied to ΔAMO, AO² = AM² + MO², supplies an equation for the radius r:

(1 - r)² = (1/2)² + (x + r)²,

which, with x = 3/5, simplifies to a linear equation in r so that r = 39/320. References

1. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989

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