In a triangle $ABC$, $D$ is the orthogonal projection of the incenter $I$ onto $BC$. Line $DI$ meets the incircle again at $E$. Line $AE$ intersects side $BC$ at point $F$. Suppose that the segment IO is parallel to $BC$, where $O$ is the circumcenter of $\triangle ABC$. If $R$ is the circumradius and $r$ the inradius of the triangle, prove that $EF=2(R-2r)$.