MHB Circles in a square and diameter of the circle

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The problem involves a larger circle inscribed in a square with sides measuring 40 units. A smaller circle, tangent to the larger circle and two sides of the square, is placed in one corner of the square. The challenge is to determine the diameter of this smaller circle. The solution requires understanding the geometric relationships between the circles and the square. The problem highlights the complexity of seemingly simple geometric configurations.
Wilmer
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A circle is inscribed in a square with sides = 40.

A smaller (of course!) circle tangent to the above
circle and 2 sides of the square is inscribed in
one of the corners of the square.

What is the diameter of this circle?
 
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$20(\sqrt2-1)$, easy.
 
No.
diameter = 40(3-2√2) = ~6.86
Not as easy as it appears...
 
You’re right. I overlooked the teeny bit in the extreme corner.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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