72 THEORY OF COLLINEATIONS. 
4). For the same reason all lines through B and C are in- 
variant lines. 
Now the intersection of two invariant lines is an invariant 
point ; hence every point in the plane is an invariant point, 
for it is the intersection of at least two invariant lines; also 
all lines are necessarily invariant lines. A transformation T 
which leaves every point of the plane invariant is an identical 
transformation. 
If the fourth invariant point D be taken on one side of the 
invariant triangle ABC, for example on BC, then the one- 
dimensional transformation along the invariant line BC leaves 
three points B, C, D invariant and therefore leaves all points 
on the line invariant. Consequently all lines through A are 
invariant lines, for each has two invariant points, one at A 
and the other at its intersection with BC. All points on a 
line through A are not invariant points of the transformation 
T, which is therefore not an identical transformation. This 
case will be discussed later. 
In the same way it may be shown that a collineation of the 
plane which leaves four lines invariant, no three of which 
pass through a point, leaves all lines and all points of the 
plane invariant and is an identical collineation. 
THEOREM 6. A plane collineation 7 which leaves invariant 
four points forming a quadrangle or four lines forming a quadri- 
lateral leaves all points and lines of the plane invariant and is an 
identical collineation. 
$2. Geometric Construction of Plane 
Collineations. 
86. Geometric Methods. Thus far in this chapter we have 
considered the plane collineation, or two-dimensional projective 
transformation, from the analytic point of view. Weshall now 
reconsider the same subject and obtain the same results by 
means of geometric construction. Each method is alone suffi- 
cient for the foundations of the theory of collineations, but 
