Hence 



PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 



Z ACE = Z BAE. 

 ^ Z ABD Z BAD 



767 



ac 



Z ACE Z CAE • 



From the definition of angle this last expression is the double ratio of 

 the lines AD, AE, AB, AF, i. e. equal to 



(DE i QF) = (BC I AQ) = (CB | AQ). 



Figure 19. 



From the definition of distance the last ratio is 



iib AG 



AC 

 AB 



Hence 



or 



ac 

 ab-AB 



AB' 

 ac • AC. 



We have already remarked that ac is uniquely determined by AC. If 

 then we keep AC constant 



ab-AB = k. 



The product of side and angle in an isosceles triangle is an absolute 

 constant. We shall choose the units so that the constant is unity. 



Then 



ah = 



AB 



(9) 



