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Part of 2021 Iran MO (2nd Round)

Problems(1)

tangency implying cyclic quadrilateral

Source: 2021 Iran second round mathematical Olympiad P3

5/15/2021
Circle ω\omega is inscribed in quadrilateral ABCDABCD and is tangent to segments BC,ADBC, AD at E,FE,F , respectively.DEDE intersects ω\omega for the second time at XX. if the circumcircle of triangle DFXDFX is tangent to lines ABAB and CDCD , prove that quadrilateral AFXCAFXC is cyclic.
geometrycircumcirclecyclic quadrilateral