$20 \cdot 109$ As shown in the figure, points $B$ and $C$ are on line segment $A D$, and $A B, B C$, and $C D$ are the diameters of circles $O, N, P$ respectively. The radii of circles $O, N, P$ are all 15. Line segment $A G$ is tangent to circle $P$ at $G$.
If $A G$ intersects circle $N$ at $E$ and $F$, then the length of chord $E F$ is
(A) 20.
(B) $15 \sqrt{2}$.
(C) 24.
(D) 25.
(E) None of the above.
[Solution] Connect $P G, N F$, and draw $N M$ perpendicular to $A G$ intersecting $A G$ at $M$.
From $\triangle A M N \backsim \triangle A G P$, we have
$$
\frac{M N}{A N}=\frac{G P}{A P} \text { or } \frac{M N}{45}=\frac{15}{75} \text {, }
$$

which gives $M N=9$.
In $\triangle M N F$, by the Pythagorean theorem, we have $(M F)^{2}=15^{2}-9^{2}=144$.
Thus, $M F=12$, so $E F=24$.
Therefore, the answer is $(C)$.