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cyclic wanted, points symmetric to intersection of perp. diagonals wrt sides

Source: 2011 All-Ukrainain Correspondence MO of magazine ''In the World of Mathematics'', grades 5-12 p11

April 26, 2021
CyclicgeometrySymmetricUkraine Correspondence

Problem Statement

In a quadrilateral ABCDABCD, the diagonals are perpendicular and intersect at the point SS. Let K,L,MK, L, M, and NN be points symmetric to SS with respect to the lines AB,BC,CDAB, BC, CD, and DADA, respectively, BNBN intersects the circumcircle of the triangle SKNSKN at point EE, and BMBM intersects circumscribed the circle of the triangle SLMSLM at the point FF. Prove that the quadrilateral EFLKEFLK is cyclic .
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