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Prove that ABCD is a rhombus

Source: Brazilian M.O. 2004

October 17, 2004
geometryrhombusconicshyperbolasymmetryinradiusincenter

Problem Statement

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