2. Given $I$ as the incenter of $\triangle A B C$, $M$ as the midpoint of side $A C$, $A I \perp I W$, and $C I$ intersects the circumcircle of $\triangle A B C$ at a second point $W$. Find $\frac{C I}{I W}$.
Extend $I W$ to point $J$, such that $W J=W I$. By Proposition 1, $J A \perp A I$.
Then $J A / / I M$.
Therefore, $I J=I C, \frac{C I}{I W}=2$.