Bunuel wrote:

Four circles of radius 2 with centers A, B, C and D are arranged symmetrically around another circle of radius 2, and four smaller equal circles with centers E, F, G and H each touch three of the larger circles as shown in the figure above. What is the radius of one of the smaller circles?

(A) √2 – 2

(B) √2 – 1

(C) 2√2 – 2

(D) 1

(E) 3√2 – 1

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All large circles have equal radii, \(r\) = 2. All small circles have equal radii.*

Find the length of the square's diagonal, then the length of the square's side.

From the square's side, subtract two radii of length 2: that is the diameter of small circle.

Divide result by 2 to get the radius of small circle.

Length of square's diagonal AC (see diagram)

Diagonal length = (radius of circle A) + (radius of middle circle * 2) + (radius of circle C)

Diagonal length = (2 + 4 + 2) = 8

Side length, \(s\), of the square with diagonal \(d\)?\(s^2 = d\)

\(s = \frac{d}{\sqrt{2}}\)

\(s = \frac{8}{\sqrt{2}} = (\frac{8}{\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}})=4\sqrt{2}\)

Side of square and diameter of small circle\(s\) = (radius of circle B) + (diameter of small circle F) + (radius of circle C) (see diagram, BC)

\(s\) = (2) + (diameter of small circle) + (2)

\(s\) = (diameter of small circle) + 4

diameter of small circle = \((s - 4) = (4\sqrt{2} - 4)\)

Small circle's radiusDiameter of small circle = (side length) - 4

Diameter of small circle =

\(4\sqrt{2} - 4\)Radius of small circle = \(\frac{1}{2}\)diameter, so

Radius of small circle =

\(\frac{4\sqrt{2} - 4}{2}\)Radius of small circle = \(2\sqrt{2} - 2\)

Answer C

*All the circles are tangent. Their common tangency points are perpendicular to their radii, thus all radii lie on the same line(s).

The radii can be summed to find diagonal and side of square.
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