Vol. 7 No 5 - Pi Mu Epsilon

THE AREA OF A TRIANGLE FORMED BY THREE LINES
notation and methods familiar to students taking a first course in
linear algebra.
We begin by forming the coefficient matrix p of the system (2)
and the matrix Q . - -
-
One way to determine a triangle is to
specify three noncollinear points X(x, x2),
y(~l, Y) and Z(z, z ) to be used as
vertices (Figure 1). It is well known [~oble,
Daniel, 1977, p. 2091 that the area, A, of the
triangle is given by the formula:
where Mi, M b and Mc in Q are cofactors of elements a bi and ci in
i
P. For example,
where the sign is chosen to make A positive.
Another way to determine a triangle is to specify three non-
current lines, no two parallel
We note that the condition that no two lines are parallel to each other
implies that the cofactors Ma , , and Me are all non-zero. Further-
3 \ 3
more, with this notation, formula (3a) becomes
la! a2 "31
2
which enclose the triangle (Figure 1). Though it is an old result
[salmon, 1879, p. 32 1, it is not so well known that the area, A, of the
triangle is also given by the formula
Is it possible for the determinant of matrix P, detP, to equal
zero If it is, there will exist a non-trivialsolution (s,,
the system
(5a)
a1s t a2s t ass3 = 0
b s t b s tb3s3=0
11 22
s2, 8,) to
The purpose of this note is to prove the formula (3a) using
If s3 # 0, then
equations in (5a).
s2/s3,1) is also a solution to the system of
Thus all three lines of system (2) pass through the