Here's a vector proof that seems to work.

Denoting the bisector of BEC as EL, the scalar product of with unit vectors in the directions of EC and EB will be equal, so,

Now,

and

,

so adding, and dividing by 2,

for some constants and

If MN and EL are to be at right angles to each other, their scalar product will have to equal zero so, using (1),

will have to be some multiple of and will have to be the same multiple of implying in turn that and are equal in magnitude.