Example 17 Given the coordinates of the vertices of $\triangle A B C$ are $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right)$, points $E$ and $F$ are on lines $A C$ and $A B$ respectively, and $\frac{\overline{A F}}{\overline{F B}}=\frac{m}{l}, \frac{\overline{A E}}{\overline{E C}}=\frac{n}{l}(m \neq-l, n \neq-l, l+m+n \neq 0)$, find the coordinates of the intersection point of lines $B E$ and $C F$.
Solving, by the section formula, the x-coordinate of point $E$ is
$$
x_{E}=\frac{x_{1}+\frac{n}{l} x_{3}}{1+\frac{n}{l}}=\frac{l x_{1}+n x_{3}}{l+n} .
$$

On the other hand, by Menelaus' theorem, we have $\frac{\overline{A F}}{\overline{F B}} \cdot \frac{\overline{B P}}{\overline{P E}} \cdot \frac{\overline{E C}}{\overline{C A}}=-1$, thus $\frac{\overline{B P}}{\overline{P E}}=\frac{l}{m} \cdot \frac{n+l}{l}=\frac{n+l}{m}$.
Therefore, using the section formula again, the x-coordinate of point $P$ is
$$
x_{P}=\frac{x_{2}+\frac{n+l}{m} x_{E}}{1+\frac{n+l}{m}}=\frac{l x_{1}+m x_{2}+n x_{3}}{l+m+n} .
$$

Similarly, the y-coordinate of $P$ is $y_{P}=\frac{l y_{1}+m y_{2}+n y_{3}}{l+m+n}$.
Therefore, the coordinates of the intersection point are $\left(\frac{l x_{1}+m x_{2}+n x_{3}}{l+m+n}, \frac{l y_{1}+m y_{2}+n y_{3}}{l+m+n}\right)$.