$17 \cdot 167$ If $D$ is a point on side $BC$ of $\triangle ABC$, and the centroids of $\triangle ABD$ and $\triangle ACD$ are $E$ and $F$ respectively, then $S_{\triangle AEF}: S_{\triangle ABC}$ equals
(A) $4: 9$.
(B) $1: 3$.
(C) $2: 9$.
(D) $1: 9$.
(China Sichuan Province Junior High School Mathematics Competition, 1994)
[Solution] As shown in the figure, extend $A E, A F$ to intersect $B C$ at $M, N$, respectively. Thus, $M, N$ are the midpoints of $B D$ and $D C$.
By the properties of the centroid of a triangle, we have:
$S_{\triangle A E F}: S_{\triangle A M N}=2^{2}: 3^{2}=4: 9$,
and $S_{\triangle A M N}: S_{\triangle A B C}=1: 2$,
therefore, $S_{\triangle A E F}: S_{\triangle A B C}=2: 9$.
Hence, the answer is $(C)$.