Every linear equation determines a line. Moreover, each such equation determines a line uniquely. For example, the given equation lx + my + nz = 0 describes coordinates of points Q(-n,0,l) and R(m,-l,0). This gives the equation of the straight line QR as follows:
ª x y zª = 0 or l(lx + my + nz) = 0.
Furthermore, this line is unique. The two points Q(-n,0,l) and R(m,-l,0) are the only points on QR whose y, z coordinates are zero, respectively.
Two straight lines having one common point can be described by the following two equations:
l1x + m1y + n1z = 0 and l2x + m2y + n2z = 0.
Solving these equations gives three ratios:
x = y = z
m1n2 - m2n1 n1l2 - n2l1 l1m2 -l2m1
Hence, "the ratios of x, y, z are the ratios of
m1n2 - m2n1, n1l2 - n2l1, l1m2 - l2m1."
Moreover, unless they all vanish, the ratios determine a unique point. That particular point lies on each of the lines. For example, given that the ratios all vanish, then the following is true:
Three straight lines having a common point can additionally be described by the equations:
l1x + m1y + n1z = 0, l2x + m2y + n2z = 0, and l3x + m3y + n3z = 0 These equations have this common point:
ªl1 m1 n1ª = 0.
ªl2 m2 n2ª
ªl3 m3 n3ª
Three lines with a common point are said to be concurrent. In contrast, a triangle may be defined as "the figure formed by three non-concurrent lines." These non-concurrent lines are called the triangle's "sides," and the three points where the sides meet are known as the triangle's "vertices."
For instance, the triangle of reference is given by the points X(1,0,0), Y(0,1,0), and Z(0,0,1). The side YZ of this triangle is defined as follows:
ªx y zª = 0
It should also be noted that t...