Prove the perpendicularity of the sides connected with the quadrilateral

Question

In the quadrangle $UEFA$, point $O$ -- is the intersection of diagonals and it is also known that $OF = OU$ and $OE = OA$. The points $S$ and $N$ are such that $ES = AE$, $FU = NU$ and $SO$ is perpendicular to $AE$ and $NO$ is perpendicular to $FU$. I want to prove that $SN$ is perpendicular to one side of $UEFA$.

I made a drawing to the problem to make it clearer. You can see that for example $SN$ is perpendicular to $AF$ or $UE$. More accurate plotting is available on GeoGebra

In what ways can we prove the perpendicularity of $SN$ to the side of the quadrilateral? I tried to consider the $UESA$ quadrangle and try to prove that $SF$ is perpendicular to $EA$, but it brought me to a standstill. Where to start proofs, is there any feature?

Answer 1

Consider the image below where $L=AF\cap SO$.

1. Demonstrate that $\triangle AOS \cong \triangle EOS$ and therefore that $\triangle AES$ is equilateral.
2. Use a similar approach to show that $\triangle UFN$ is equilateral.
3. Use 1. and 2. to show that $\frac{\overline{SO}}{\overline{NO}}= \frac{\overline{AO}}{\overline{OF}}$.
4. Show that $\angle SON \cong \angle AOF$, by expressing both of them as $\frac{\pi}2 + \angle SOF$.
5. By 3. and 4., and by ASA criterion, $\triangle SON \sim \triangle AOF$. In particular, $\angle OSN \cong \angle FAO$.
6. Since $\angle ALO$ and $\angle FAO$ are complementary, and $\angle ALO \cong \angle SLF$ (vertical angles), so are $\angle SLF$ and $\angle FAO$.
7. Derive the thesis by 5. and 6.