70 THEORY OF COLLINEATIONS. 
Eliminating y we get 
6!” aa + c!/ = b/ a= ax = c! 
b/a—b alx?+(c/—-a)x—ec 0 =(0, (6) 
0 b’« —b ax? + (c’ —‘a) a —e 
When the determinant is expanded, the coefficient of x van- 
ishes and we have a cubic equation of the form, 
av + Bar'+yr+d=0, (7) 
from which to find «. Let the three roots of this cubic be 
A, A’, A”. Substituting their values in the first of equations 
(5), we find three values of y, viz.: B, B’,B”’. The three 
points whose coordinates are (A, B), (A’, B’), (A”, B”) are 
invariant points of the transformation (1). In the most gen- 
eral case these points form a triangle which is called the 
invariant triangle of the transformation. There are special 
cases to be considered when the three invariant points do not 
form a triangle; for example, two of the three points may 
coincide, or all three may lie on a line, ete. All these special 
cases will be determined later. 
THEOREM 4. A collineation of the most general kind leaves 
three linearly independent points of the plane invariant. 
84. Invariant Lines of a Collineation. If equations (1) 
be interpreted in line coordinates instead of point coordinates, 
then the analytic work of the last article shows that a colline- 
ation in a plane leaves three lines invariant; these generally 
form a triangle. We have just shown that a collineation of 
the most general kind leaves invariant three points and three 
lines. The relation of these three points and three lines is 
evident at once. The three points are the vertices and the 
three lines are the sides of the same invariant triangle. Let 
A, B, C designate the vertices, and a, b, ¢ the opposite sides 
respectively of the triangle. 
If a collineation leave both A and B invariant, then c, their 
join, is transformed into itself; for c is transformed into some 
line c,, which must pass through A and B, because they are 
unaltered in position; and hence ¢ and ¢, must be the same 
