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.//
