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March 21, 2020

Challenge Problem 10: Radius Of Circle Between Quarter-Circles

In a square with a side length of 1, two quarter circles are drawn and a circle is inscribed between the quarter circles, as shown in the diagram. What is the radius of the inscribed circle?




Connect the each of two corners of the square to the opposite tangent point in the inscribed circle, as shown below in the line segments AG and DF.


We will first show that O is the center of the inscribed circle.

From point G draw the tangent line to both the inscribed circle and the quarter circle. The perpendicular to this line at G will pass through the centers of both circles. This is point A for the quarter circle and it will contain the center O of the inscribed circle.

Similarly, from point F draw the tangent line to both the inscribed circle and the quarter circle. The perpendicular to this line at F will pass through the centers of both circles. This is point D for the quarter circle and it will contain the center O of the inscribed circle.


Since the center of the inscribed circle lies along both AG and DF, the intersection of the line segments O is the center of the inscribed circle.

Now drop a perpendicular from O to the side of the square AD and label the tangent point P.


Since AG is a radius of the quarter circle, it has a length of 1. Since OG is a radius of the inscribed circle, it has a length of r.

Thus AO = AG – OG = 1 – r.

Furthermore OP = r, and OP will bisect the side of the square so AP = 0.5.

Thus APO is a right triangle with legs 0.5 and r and hypotenuse 1 – r. By the Pythagorean Theorem:

0.5^2 + r^2 = (1 – r)^2
0.5^2 + r^2 = 1 – 2r – r^2
0.5^2 = 1 – 2r
2r = 1 – 0.5^2
r = 0.75/2
r = 0.375 = 3/8

References

Pradeep Hebbar puzzle on Quora
https://qr.ae/TZFy7x

Math StackExchange help
https://math.stackexchange.com/questions/3486090/circle-inscribed-between-quarter-circles-proving-its-center-point

Mind Your Decision

Yêu Tiếng Anh

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