11. Let $D$ be a point inside acute $\triangle A B C$ such that $\angle A D B=\angle A C B+90^{\circ}$, and $A C \cdot B D=A D \cdot B C$. Calculate the ratio $\frac{A B \cdot C D}{A C \cdot B D}$.
11. Let the origin of the complex plane be $C$, and the line $CA$ be the real axis. Suppose points $A$, $B$, and $D$ correspond to the complex numbers $1$, $z$, and $d$, respectively. Then $\frac{BC}{AC} = \frac{BD}{AD}$. Therefore, 
$$
\frac{\overrightarrow{CB}}{\overrightarrow{CA}} = z, \frac{\overrightarrow{DB}}{\overrightarrow{DA}} = zi, \overrightarrow{DA} = 1 - d, \overrightarrow{DB} = z - d,
$$
so
$$
d = \frac{z(i-1)}{zi-1},
$$
and thus
$$
\frac{|AB| \cdot |CD|}{|AC| \cdot |BD|} = \sqrt{2}.
$$