6. Let $D$ be a point on side $BC$ of $\triangle ABC$, and points $E, F$ be the centroids of $\triangle ABD$ and $\triangle ACD$, respectively. Connecting $EF$ intersects $AD$ at point $G$. What is the value of $\frac{DG}{GA}$?
(1991-1992 Guangzhou and Four Other Cities Competition Question)
6. Connect $B E, C F$, and extend them to intersect at $M$, then $M$ is the midpoint of $A D$. Since $E, F$ are the centroids of $\triangle A B D$ and $\triangle A C D$ respectively, then $\frac{M E}{M B}=\frac{M F}{M C}=\frac{1}{3}$. Therefore, $E F / / B C, E G / / B D$. Thus $\frac{M G}{D M}=\frac{M E}{M B}=\frac{1}{3}, \frac{D G}{D M}=\frac{B E}{B M}=\frac{2}{3}, M G=\frac{1}{3} D M, D G=$ $\frac{2}{3} D M, A G=A M+M G=D M+\frac{1}{3} D M=\frac{4}{3} D M$, hence $D G: G A=\frac{2}{3} D M: \frac{4}{3} D M=1: 2$.