I'm trying to solve this using elementary geometry.

Let $\triangle ABC$ be a triangle and consider the squares $CBFG$ and $ACDE$. Draw $\overline{EF}$ and let $M$ be its midpoint. Draw the perpendicular to $\overline{AB}$ passing through $M$. Let $N$ be the intersection point of this perpendicular and $\overline{AB}$ . Prove that $N$ is the midpoint of $\overline{AB}$.

[![image of the geometry problem](https://i.stack.imgur.com/Yun4P.png)](https://i.stack.imgur.com/Yun4P.png)

Any ideas?

I've tried extending $MA$ to $A'$, with $MA=MA'$, same with $MB$. I draw $A'F$, $B'E$, expecting them to be congruent to $ABC$, but I couldn't prove it (they should be congruent). A lot of lines here and there, but I couldn't say anything relevant.

I cannot use complex numbers. The tools allowed are parallelism, similarity, congruences.