［Solution］From the given equation with equal roots, we have
$$
\Delta=(-8)^{2}-4(m+6)=0 \text {, which gives } m=10 \text {. }
$$

Thus, the equation becomes $x^{2}-8 x+16=0$.
Let $A E$ be the altitude of $\triangle A B C$, and $A D$ be the median of $B C$ (as shown in the figure).
Also, $B C=x_{1}=x_{2}=4$, and $\frac{1}{2} B C \cdot A E=6$, so $A E=3$.
By the centroid theorem, $G F: A E=G D: A D=1: 3$,
i.e., $G F: 3=1: 3, \therefore G F=1$.
Therefore, the answer is $(B)$.
17.91 Given that the equation $x^{2}-8 x+m+6=0$ has two equal real roots, and its root is the length of side $B C$ of $\triangle A B C$. If the area of $\triangle A B C$ is 6, then the distance from the centroid $G$ to $B C$ is
(A) 2.
(B) 1.
(C) 3.
(D) $1 \frac{1}{2}$.
(China Zhejiang Province Junior High School Mathematics Competition, 1992)