Problem 1093. Square, Circle, Tangent, Radius, Side, Sangaku, Japanese

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The figure below shows a square ABCD of side 1 and center O. Equal circles O₁ and O₂ are inscribed in the square and tangent at O. EF and E₁F₁ are common tangents to O₁ and O₂. Circle O₃ of radius x is tangent to AB, BC, and EF. Circle O₄ of radius x is tangent to AD, CD, and E₁F₁, Find x.
 

Reference: Fukagawa Hidetoshi, Tony Rothman, Sacred Mathematics: Japanese Temple Geometry (Princenton University Press, 2008).

See also:
Geometric art problem 1093