Let

(1)

If the lines $L_1 = 0$ and $L_2 = 0$ have an intersection point $P$ , then, by the parent entry, the equation

(2)

Theorem. A necessary and sufficient condition in order to three lines $$L_1 = 0, \quad L_2 = 0, \quad L_3 = 0$$ pass through a same point, is that the determinant formed by the coefficients of their equations (1) vanishes: $$ \left|\begin{matrix} A_1 & B_1 & C_1\\ A_2 & B_2 & C_2\\ A_3 & B_3 & C_3 \end{matrix}\right| = \left|\begin{matrix} A_1 & A_2 & A_3\\ B_1 & B_2 & B_3\\ C_1 & C_2 & C_3 \end{matrix}\right| = 0. $$

Proof. If the line $L_3 = 0$ belongs to the fan of lines determined by the lines $L_1 = 0$ and $L_2 = 0$ , i.e. all the three lines have a common point, there must be the identity $$L_3 \equiv L_1+L_2,$$ i.e. there exist three real numbers $k_1$ , $k_2$ , $k_3$ , which are not all zeroes, such that the equation

(3)

Suppose conversely that the determinant vanishes. This implies that the above group of equations has a nontrivial solution $k_1,\,k_2,\,k_3$ . Thus we can write the identic equation (3). Let e.g. $k_1 \neq 0$ . Solving (3) for $L_1$ yields $$L_1 \equiv -\frac{k_2L_2+k_3L_3}{k_1},$$ which shows that the line $L_1 = 0$ belongs to the fan determined by the lines $L_2 = 0$ and $L_3 = 0$ ; so the lines pass through a common point.

Bibliography

1

LAURI PIMIÄ: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).