Solution:
It is given the centroid is origin and two vertices are $(2,4,6)$ and $(0,-2,-5)$
Suppose the third vertex be $(x, y, z)$
The coordinates of the centroid for a triangle is given by the average of the coordinates of its vertices.
$\begin{array}{l}
\Rightarrow(0,0,0)=\left(\frac{2+0+x}{3}, \frac{4+(-2)+y}{3}, \frac{6+(-5)+z}{3}\right) \\
\Rightarrow \frac{2+x}{3}=0, \therefore x=-2 \\
\Rightarrow \frac{2+y}{3}=0, \therefore y=-2 \\
\Rightarrow \frac{1+x}{3}=0, \therefore x=-1
\end{array}$
So, the third vertex is $(-2,-2,-1)$.