41. Under what condition do three lines, given by the equations: $A_{1} x+B_{1} y+C_{1}=0 ; \quad A_{2} x+B_{2} y+C_{2}=0 \quad$ and $\quad A_{3} x+B_{3} y+C_{3}=0$, pass through the same point?
41. Finding the coordinates of the intersection point of the second and third line (see problem 39), we get the following formulas:

$$
x=\frac{\left|\begin{array}{ll}
B_{2} & C_{2} \\
B_{3} & C_{3}
\end{array}\right|}{\left|\begin{array}{ll}
A_{2} & B_{2} \\
A_{3} & B_{3}
\end{array}\right|} ; \quad y=\frac{\left|\begin{array}{cc}
C_{2} & A_{2} \\
C_{3} & A_{3}
\end{array}\right|}{\left|\begin{array}{ll}
A_{2} & B_{2} \\
A_{3} & B_{3}
\end{array}\right|}
$$

If we want the first line to pass through this point as well, the coordinates of the point must satisfy the first equation, i.e.,

$$
A_{1} \cdot \frac{\left|\begin{array}{ll}
B_{2} & C_{2} \\
B_{3} & C_{3}
\end{array}\right|}{\left|\begin{array}{ll}
A_{2} & B_{2} \\
A_{3} & B_{3}
\end{array}\right|}+B_{1} \cdot \frac{\left|\begin{array}{cc}
C_{2} & A_{2} \\
C_{3} & A_{3}
\end{array}\right|}{\left|\begin{array}{ll}
A_{2} & B_{2} \\
A_{3} & B_{3}
\end{array}\right|}+C_{1}=0
$$

Otherwise:

$$
A_{1}\left|\begin{array}{ll}
B_{2} & C_{2} \\
B_{3} & C_{3}
\end{array}\right|+B_{1}\left|\begin{array}{ll}
C_{2} & A_{2} \\
C_{3} & A_{3}
\end{array}\right|+C_{1}\left|\begin{array}{ll}
A_{2} & B_{2} \\
A_{3} & B_{3}
\end{array}\right|=0
$$

The expression on the left side can be written as a determinant of the third order

$$
\left|\begin{array}{lll}
A_{1} & B_{1} & C_{1} \\
A_{2} & B_{2} & C_{2} \\
A_{3} & B_{3} & C_{3}
\end{array}\right|
$$

Thus, three lines pass through the same point if and only if

$$
\left|\begin{array}{lll}
A_{1} & B_{1} & C_{1} \\
A_{2} & B_{2} & C_{2} \\
A_{3} & B_{3} & C_{3}
\end{array}\right|=0
$$