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Peruvian geometry

Source: Peru CSMO TST 2019(Ukraine 2017)

March 25, 2019
geometry

Problem Statement

Let ABAB be a diameter of a circle Γ\Gamma with center OO. Let CDCD be a chord where CDCD is perpendicular to ABAB, and EE is the midpoint of COCO. The line AEAE cuts Γ\Gamma in the point FF, the segment BCBC cuts AFAF and DFDF in MM and NN, respectively. The circumcircle of DMNDMN intersects Γ\Gamma in the point KK. Prove that KM=MBKM=MB.
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