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3 cyclic quadrilaterals with equidistant centres

Source: Sharygin 2011 Round 2 grade 9 p4

July 2, 2018
geometrycyclic quadrilateral

Problem Statement

Quadrilateral ABCDABCD is inscribed into a circle with center OO. The bisectors of its angles form a cyclic quadrilateral with circumcenter II, and its external bisectors form a cyclic quadrilateral with circumcenter JJ. Prove that OO is the midpoint of IJIJ.
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