Direct argument for Nash equilibrium
(m, m) is an equilibrium: if either candidate chooses a different position she loses.
No other pair of positions is a Nash equilibrium:
if one candidate loses then she can do better by moving to m (where she either wins outright or ties for first place)
if the candidates tie (because their positions are either the same or symmetric about m), then either candidate can do better by moving to m, where she wins outright.