Question 58
Let $ABC$ be an acute triangle with circumcenter $O$ and let $D$ be the foot of the perpendicular from $A$ to $BC$. Assume $\angle AOD = 90^\circ$ and $OD = 4\sqrt{7}$ hold. Let $E$ and $F$ be the feet of perpendiculars from $D$ to $AB$ and $AC$ respectively, and let the lines $AO$ and $EF$ meet at $P$. If $AP = 11$, find the length of $EF$.