<MNO=<ONG (given)
GO=MO=EO=DO (radii)
<GON=<MON (symmetry of the bisector)
<OEG=<OGE (isosceles triangle OEG formed by radii GO and EO)
Similarly <OMD=<ODM.
∆s NOD and NOE are congruent as are GOE and MOD, so the chords MD and GE have equal length.
The question doesn’t ask us to find or prove anything.