BEC is a right triangle because be is perpendicular to AC.

And because m is the midpoint of BC, EM=BM=CM.

Similarly, CF is perpendicular to AB, CFB is a right triangle, and FM=BM=CM.

So EM=FM, EFM is an isosceles triangle.

Because n is the midpoint of EF, MN is perpendicular to EF (isosceles triangle with three lines in one)