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Cyclic quads wanted

Source: 2007 Bulgarian Autumn Math Competition, Problem 9.3

March 17, 2022
geometrycyclic quadrilateral

Problem Statement

Let the intersection of the diagonals ACAC and BDBD of the convex quadrilateral ABCDABCD be point EE. Let MM be the midpoint of AEAE and NN be the midpoint of CDCD. It's known that BDBD bisects ∠ABC\angle ABC. Prove that ABCDABCD is cyclic if and only if MBCNMBCN is cyclic.
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