You should use Thales theorem (not the one about right triangle in circle, but the one about parallel line to a triangle side).BM, CN and DG are all parallels each other, because they're all perpendicular to a same straight line.As D is BC mid point, we have BD = DCor we can also say (BC/BD) = 2 and (BC/DC) = 2And thus, if BD = DC with BM // DG // CN and with M, G and N on the same lineWe have a ratio conservation ; that means (MN/MG) = 2 and also (MN/GN) = 2thus we find that MG = GN;;;

en.wikipedia.org