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Concurrent lines through midpoints in a quadrilateral

Source: Czech-Polish-Slovak Match, 2011

August 9, 2011
symmetrygeometrycyclic quadrilateralgeometry unsolved

Problem Statement

In convex quadrilateral ABCDABCD, let MM and NN denote the midpoints of sides ADAD and BCBC, respectively. On sides ABAB and CDCD are points KK and LL, respectively, such that ∠MKA=∠NLC\angle MKA=\angle NLC. Prove that if lines BDBD, KMKM, and LNLN are concurrent, then ∠KMN=∠BDCand∠LNM=∠ABD. \angle KMN = \angle BDC\qquad\text{and}\qquad\angle LNM=\angle ABD.
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