Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

46

ANALYTIC GEOMETRY

THEOREM 3. The lines L1 and L2 are parallel * if and only if
A1: A2 = B1: B2
For, L1 and L2 are parallel if and only if the angle cp between them, as defined in ~ 7, is zero; but, according to ~ 7,
Ex. 12, <, or better, tan f, is zero, when and only when
A1B2 - AB1 = 0. But this equation is equivalent to the proportion A: A2 = B: B2.
As a second consequence of ~ 7, Ex. 12, we obtain the following theorem.
THEOREM 4. The lines L1 and L2 are perpendicular if and
only if
A1A2 + B1B2 = 0.
Identical Lines. Two equations do not have to be identically
the same in order to represent the same line. For example,
the equations,
2 x - y + 1 = 0,
6x -3y +3=0,
represent the same line. The corresponding constants in
them are not equal, but they are proportional. We have,
namely,
2:6=-1: -3=1:3,
or, what amounts to the same thing,
2: -1:1=6: -3:3.
This condition is general. We formulate it as a theorem:
THEOREM 5. The lines L1 and L2 are identical if and only if
A1:A2=B:B2 = Cl: C2,
or A: B: C0 =A2: B2: C2.
For, L1 and L2 are the same line when and only when they
have the same slope and the same intercept on the axis of y,
that is, when and only when
-A1 A2 and C C2
B1 B2 B1 B2'
* Or, in a single case, identical. Cf. Th. 5.