Example 2. In
$\triangle \mathrm{ABC}$, it is
known that the coordinates of
$\mathrm{B}$ and $\mathrm{C}$ are
(0, 0) and (a,
0), respectively. When $\mathrm{A}$ moves
on the line $l$:
$x+2 y-8=0$, find the equation of the locus of the centroid of $\triangle A B C$.
Analysis: As shown in Figure 2, $P(x, y)$ is the centroid of $\triangle ABC$. Clearly, the movement of $P$ is restricted by $A$. If $M$ is the midpoint of $BC \left(\frac{a}{2}, 0\right)$, and $A$ is $\left(x_{1}, y_{1}\right)$, then $|AP|: |PM| = 2: 1$. By the section formula, we get: $x_{1}=3x-a, y_{1}=3y$. Using the fact that $A$ lies on the line $x_{1}+2y_{1}-8=0$, substituting the above relations, we get: $3x-a+2 \times 3y-8=0$. Therefore, $3x+6y-a-8=0$ is the equation of the locus of the centroid $P$.