Example 21 Given three lines $l_{1}, l_{2}, l_{3}$ on a plane that do not pass through the origin, their equations are:
$$
\begin{array}{l}
l_{1}: a_{1} x+b_{1} y+c_{1}=0, \\
\iota_{2}: a_{2} x+b_{2} y+c_{2}=0, \\
\iota_{3}: a_{3} x+b_{3} y+c_{3}=0 .
\end{array}
$$

A moving line $l$ passing through the origin intersects the lines $l_{1}, l_{2}, l_{3}$ at points $A\left(x_{1}, y_{1}\right), B\left(x_{2}\right.$, $\left.y_{2}\right), C\left(x_{3}, y_{3}\right)$. Let $\left|\overrightarrow{O A}_{i}\right|=\rho_{i}(i=1,2,3),$ and $P$ be a point on the line $l$, with $|\overrightarrow{O P}|=$ $\rho$, and satisfies: $\frac{1}{\rho_{1}}+\frac{1}{\rho_{2}}+\frac{1}{\rho_{3}}=\frac{1}{\rho}$. Try to find the equation of the trajectory of the moving point $P(x, y)$.
Let the inclination angle of line $l$ with the $x$-axis be $\theta$, then we have
$$
\left\{\begin{array}{l}
x = \rho \cos \theta, \\
y = \rho \sin \theta,
\end{array} \quad \left\{\begin{array}{l}
x_{i}=\rho_{i} \cos \theta, \\
y_{i}=\rho_{i} \sin \theta.
\end{array} \quad (i=1,2,3)\right.\right.
$$

Since $A_{i}$ lies on line $l_{i}$, then $a_{i} x_{i} + b_{i} y_{i} + c_{i} = 0$.
That is, $a_{i} \rho_{i} \cos \theta + b_{i} \rho_{i} \sin \theta + c_{i} = 0$.
Since $c_{i} \neq 0$, we have $\frac{1}{\rho_{i}} = -\frac{a_{i} \cos \theta + b_{i} \sin \theta}{c_{i}} \quad (i=1,2,3)$.
Adding the three equations and noting that
$$
\frac{1}{\rho} = \frac{1}{\rho_{1}} + \frac{1}{\rho_{2}} + \frac{1}{\rho_{3}}, \quad \left\{\begin{array}{l}
x = \rho \cos \theta, \\
y = \rho \sin \theta,
\end{array}\right.
$$

we get $\frac{1}{\rho} = \frac{1}{\rho_{1}} + \frac{1}{\rho_{2}} + \frac{1}{\rho_{3}} = -\left(\frac{a_{1} \cos \theta + b_{1} \sin \theta}{c_{1}} + \frac{a_{2} \cos \theta + b_{2} \sin \theta}{c_{2}} + \frac{a_{3} \cos \theta + b_{3} \sin \theta}{c_{3}}\right)$
$$
= -\left(\frac{a_{1}}{c_{1}} + \frac{a_{2}}{c_{2}} + \frac{a_{3}}{c_{3}}\right) \cos \theta - \left(\frac{b_{1}}{c_{1}} + \frac{b_{2}}{c_{2}} + \frac{b_{3}}{c_{3}}\right) \sin \theta.
$$

From this, we can see that the equation of the locus of point $P$ is $\left(\frac{a_{1}}{c_{1}} + \frac{a_{2}}{c_{2}} + \frac{a_{3}}{c_{3}}\right) x + \left(\frac{b_{1}}{c_{1}} + \frac{b_{2}}{c_{2}} + \frac{b_{3}}{c_{3}}\right) y + 1 = 0$ (which is a straight line).