The figure shows a
cyclic
quadrilateral ABCD. E, F, G, and H are the
incenters and ra,
rb, rc, and rd are the
inradii
of the triangles ABD, ABC, BCD and ACD respectively. Prove that
EFGH is a rectangle and ra + rc = rb
+ rd. See the
proof.

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