Go Geometry Problem Problem 336. Two equal circles, a Common Tangent and a Square

The figure shows circles A (radius r) and B (radius r) tangent at C. A common tangent line is tangent to the circles at C and D, respectively. Points F and G are on DE, M is on arc CD and H is on arc CE. If FGHM is a square of side x, prove that Conclusion to prove.

Tangent circles and square

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

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