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ABCD IS tangential, if KLMN is cyclic then ABCD is cyclic

Source: Iranian Geometry Olympiad 2018 IGO Advanced p4

September 19, 2018
tangential quadrilateralgeometrycyclic quadrilateralangle bisector

Problem Statement

Quadrilateral ABCDABCD is circumscribed around a circle. Diagonals AC,BDAC,BD are not perpendicular to each other. The angle bisectors of angles between these diagonals, intersect the segments AB,BC,CDAB,BC,CD and DADA at points K,L,MK,L,M and NN. Given that KLMNKLMN is cyclic, prove that so is ABCDABCD.
Proposed by Nikolai Beluhov (Bulgaria)
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