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cyclic wanted, ABCD with _|_ diagonals related, reflections of intersections

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 C2 p1

September 30, 2021
geometryreflectionCyclicConcyclic

Problem Statement

Let ABCDABCD be a quadrilateral whose diagonals are perpendicular, and let SS be the intersection of those diagonals. Let K,L,MK, L, M and NN be the reflections of SS on the sides ABAB, BCBC, CDCD and DADA respectively. BNBN cuts the circumcircle of △SKN\vartriangle SKN at EE and BMBM cuts the circumcircle of △SLM\vartriangle SLM at FF. Prove that the quadrilateral EFLKEFLK is cyclic.
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