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4 incircles in convex ABCD concurrent if ABCD is rhombus (2013 Kyiv City MO 8.5)

Source:

August 4, 2020
geometryrhombusconcurrentcirclesconcurrency

Problem Statement

Let ABCDABCD be a convex quadrilateral. Prove that the circles inscribed in the triangles ABCABC, BCDBCD, CDACDA and DABDAB have a common point if and only if ABCDABCD is a rhombus.
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