18. (12 points) As shown in Figure 4, in the triangular prism $A B C-A_{1} B_{1} C_{1}$, it is known that $A A_{1} \perp$ the base $A B C$, $A B \perp A C$, $A C = A B = A A_{1}$, $E$ and $F$ are the midpoints of edges $B C$ and $A_{1} A$ respectively, $G$ is a point on edge $C C_{1}$, and $C_{1} F \parallel$ plane $A E G$.
(1) Find the value of $\frac{C G}{C C_{1}}$;
(2) Prove that $E G \perp A_{1} C$;
(3) Find the cosine value of the dihedral angle $A_{1}-A G-E$.
18. (1) Since $C_{1} F / /$ plane $A E G$, and
$C_{1} F \subset$ plane $A C C_{1} A_{1}$,
plane $A C C_{1} A_{1} \cap$ plane $A E G=A G$,
therefore, $C_{1} F / / A G$.
Also, since $F$ is the midpoint of $A A_{1}$, and the side face $A C C_{1} A_{1}$ is a parallelogram, then $G$ is the midpoint of $C C_{1}$, i.e., $\frac{C G}{C C_{1}}=\frac{1}{2}$.
(2) Since $A A_{1} \perp$ the base plane $A B C$, therefore,
$A A_{1} \perp A B, A A_{1} \perp A C$.
Also, $A B \perp A C$, with $A$ as the

origin, establish a spatial rectangular coordinate
system $A-x y z$, as shown in Figure 8.
Let $A B=2$.
Then, from $A B=A C=A A_{1}$ we get
$C(2,0,0), B(0,2,0)$,
$C_{1}(2,0,2), A_{1}(0,0,2)$.
Also, since $E$ and $G$ are the midpoints of $B C$ and
$C C_{1}$ respectively, then
$$
\begin{array}{l}
E(1,1,0), G(2,0,1) \\
\Rightarrow \overrightarrow{E G} \cdot \overrightarrow{C A_{1}}=(1,-1,1) \cdot(-2,0,2)=0 \\
\Rightarrow \overrightarrow{E G} \perp \overrightarrow{C A_{1}} \Rightarrow E G \perp A_{1} C .
\end{array}
$$
(3) Let the normal vector of plane $A E G$ be $\boldsymbol{n}=(x, y, z)$.
$$
\text { Then }\left\{\begin{array} { l } 
{ \boldsymbol { n } \cdot \vec { A E } = 0 , } \\
{ \boldsymbol { n } \cdot \vec { A G } = 0 }
\end{array} \Rightarrow \left\{\begin{array}{l}
x+y=0, \\
2 x+z=0 .
\end{array}\right.\right.
$$

Let $x=1$. Then $y=-1, z=-2$.
Therefore, $\boldsymbol{n}=(1,-1,-2)$.
By the problem, the normal vector of plane $A_{1} A G$ is $\boldsymbol{m}=(0,1,0)$.
Thus, $\cos \langle\boldsymbol{n}, \boldsymbol{m}\rangle=\frac{\boldsymbol{n} \cdot \boldsymbol{m}}{|\boldsymbol{n}||\boldsymbol{m}|}=-\frac{\sqrt{6}}{6}$.
Since the dihedral angle $A_{1}-A G-E$ is an obtuse angle, its cosine value is $-\frac{\sqrt{6}}{6}$.