GEOMETRIC CONSTRUCTION. 85 
THEOREM 14. The collineation 7, which transforms S into S;, 
and S; into S., transforms the conic K, determined by the projective 
pencils through S and S;,, into the conic K;, determined by the pro- 
jective pencils through S,and S.; every pair of corresponding points 
on K and K;, are collinear with S;. 
102. Construction of a Collineation by Means of K and K,. 
By making use of the principle that corresponding points on 
K and K, are collinear with S,, we can construct the line g, 
which corresponds to g, any line of the plane. 
The line g cuts K in two points, Q and R; join Q and R to 
S,; these joins cut K, in R, and S,, corresponding points to R 
and S; the line joining R, and S, is the line g, which corre- 
sponds to g. 
The transformation T transforms a point P into P,; if P be 
given, we may find P, by drawing any two lines g and g’ 
through P cutting K; find by the above construction the cor- 
responding lines g, and g,’; these intersect in P,, the point 
which corresponds to P. 
If a line g does not intersect K, the construction of g, may 
be accomplished by choosing two points, G and G’, on g and 
constructing their corresponding points, G, and G,’; these 
two new points determine g,. 
If gis a tangent to K, g, will be a tangent to K, and the 
points of contact will be collinear with S,. If the given line 
g passes through S, and cuts K in P and K, in P,, the corre- 
sponding line is found by joining P, and S,. 
THEOREM 15. A collineation of the plane can be constructed by 
means of two conies K and K; intersecting in a point S;. 
103. Invariant Points and Lines. The conics K and K, in- 
tersect in S, and generally in three other points A, B, C. 
Since any line through S, cuts K and K, ina pair of corre- 
sponding points, it follows that A, B, and C are self-corre- 
sponding points on K and K,. In other words, A, B, and C 
are invariant points of the collineation 7. The lines AB, 
BC, and CA are self-corresponding or invariant lines of the 
