In triangle $\triangle ABC$, angle $A=50^\circ$ , angle $C=65^\circ$ . Point $F$ is on $AC$ such that, $BF$ is perpendicular to $A$C. $D$ is a point on $BF$ (extended) such that $AD=AB$. E is a point $CD$ such that, AE is perpendicular to $CD$. If $BC=12$, what is the length of $EF$?
I tried and proved that $ABF \cong AFD$ and $BCF \cong CFD$. I am not able to find any relation of $EF$ with other sides.