Problem 1092. Equilateral Triangle, Square, Circle, Tangent, 90 Degree, Sangaku

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An equilateral triangle DEF, a square EGHM and a circle O touch each other in a right triangle ABC (see the figure below). Prove that AE = BC - AB
 

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