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Sangaku Geometry Theorem: Cyclic Quadrilateral

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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