100 THEORY OF COLLINEATIONS. 
Let g and g,, Fig. 17, be a pair of corresponding lines in the 
transformation T; let PP,, NN,, and MM,, be the pairs of 
points of intersection of g and g, with the sides of the in- 
variant triangle. Since k,=(ABM,M), k,=(BCP,P), and 
k,=(CAN,N) (observe the order in which the points are 
taken) we have 
Va Hd Pee AM.BM BPi.CP CM.AN_ 
Cte ANION Ii | 0B) GPa iO NIRAUNG 
But by the theorem of Menelaus* we have 
AM. BP. CN _4 ,.g AM. BP. CN _ 
BM. CP. AN oe Ain, Cle LNA 
Hence lealcek ey alla 
THEOREM 19. Every collineation of type I in the plane deter- 
mines a characteristic cross-ratio alone each of the invariant lines 
and through each of the invariant points. When these three cross- 
ratios are reckoned in the same order around the triangle their pro- 
duet is unity. 
Fic. 18. 
* Cremona, Elements of Projective Geometry, page 112. 
