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cyclic ABCD and equal angles given, perpendicular wanted

Source: Czech-Polish-Slovak Match Junior 2019, team p6 CPSJ

January 20, 2020
cyclic quadrilateralperpendiculargeometry

Problem Statement

Given is a cyclic quadrilateral ABCDABCD. Points K,L,M,NK, L, M, N lying on sides AB,BC,CD,DAAB, BC, CD, DA, respectively, satisfy ∠ADK=∠BCK\angle ADK=\angle BCK, ∠BAL=∠CDL\angle BAL=\angle CDL, ∠CBM=∠DAM\angle CBM =\angle DAM, ∠DCN=∠ABN\angle DCN =\angle ABN. Prove that lines KMKM and LNLN are perpendicular.
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