Let $ABC$ be an acute triangle inscribed on the circumference $\Gamma$. Let $D$ and $E$ be points on $\Gamma$ such that $AD$ is perpendicular to $BC$ and $AE$ is diameter. Let $F$ be the intersection between $AE$ and $BC$. Prove that, if $\angle DAC = 2 \angle DAB$, then $DE = CF$.