Solution to Exercise 3.18.
My method to overcome a difficulty is to go round it..
—George Polyá (1887–1985)

Exercise 3.18. Prove Proposition 3.2. Three distinct lines l, m, and n are all concurrent or all parallel if and only if the determinant

 

Proof. Assume three distinct lines l, m, and n are concurrent. Let (x₁, x₂, 1) be the point of concurrency. Hence,

, ,

if and only if

x₁l₁+ x₂l₂+ l₃ = 0
x₁m₁+ x₂m₂+ m₃ = 0

x₁n₁+ x₂n₂+ n₃ = 0

if and only if

The homogeneous equation has a nontrivial solution (x₁, x₂, 1); therefore,

        Next, consider the converse. Assume  Then  has a nontrivial solution (a, b, c). Note that if c is nonzero, then  is also a solution, which is a point of concurrency of the three lines by the first paragraph of the proof. Hence, assume c = 0. Thus,

 implies

Similar to the proof of the first paragraph, the first pair of equations has a nontrivial solution provided  Hence, l₁m₂ – l₂m₁ = 0. If l₁ = 0, then l₂m₁ = 0. Since l₁ and l₂ cannot both be zero, m₁ = 0. Thus, lines l[0, l₂, l₃] and m[0, m₂, m₃] are parallel. Apply a similar argument for l₂ = 0. Hence, assume l₁ and l₂ are nonzero. Thus,  Hence, m₁ = k₁l₁ and m₂ = k₁l₂. Therefore, lines l and m are parallel. A similar argument may be used for the lines l and n.//