33. [17] In quadrilateral $A B C D$, there exists a point $E$ on segment $A D$ such that $\frac{A E}{E D}=\frac{1}{9}$ and $\angle B E C$ is a right angle. Additionally, the area of triangle $C E D$ is 27 times more than the area of triangle $A E B$. If $\angle E B C=\angle E A B, \angle E C B=\angle E D C$, and $B C=6$, compute the value of $A D^{2}$.
Answer:
320
Solution:
Extend sides $A B$ and $C D$ to intersect at point $F$. The angle conditions yield $\triangle B E C \sim \triangle A F D$, so $\angle A F D=90^{\circ}$. Therefore, since $\angle B F C$ and $\angle B E C$ are both right angles, quadrilateral $E B F C$ is cyclic and
$$
\angle E F C=\angle E B C=90^{\circ}-\angle E C B=90^{\circ}-\angle E D F,
$$
implying that $E F \perp A D$.
Since $A F D$ is a right triangle, we have $\left(\frac{F A}{F D}\right)^{2}=\frac{A E}{E D}=\frac{1}{9}$, so $\frac{F A}{F D}=\frac{1}{3}$. Therefore $\frac{E B}{E C}=\frac{1}{3}$. Since the area of $C E D$ is 27 times more than the area of $A E B, E D=9 \cdot E A$, and $E C=3 \cdot E B$, we get that $\angle D E C=\angle A E B=45^{\circ}$. Since $B E C F$ is cyclic, we obtain $\angle F B C=\angle F C B=45^{\circ}$, so $F B=F C$.
Since $B C=6$, we get $F B=F C=3 \sqrt{2}$. From $\triangle E A B \sim \triangle E F C$ we find $A B=\frac{1}{3} F C=\sqrt{2}$, so $F A=4 \sqrt{2}$. Similarly, $F D=12 \sqrt{2}$. It follows that $A D^{2}=F A^{2}+F D^{2}=320$.