PHILUPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 775 



or since triangles with the same vertical angle have areas proportional 

 to the product of the including sides. 



Rearranging 



OA'C OB'A OC'B 



OAC OBA' OCB 



OA^C OB^A OC^B 

 OBA' ■ OCB' ■ OAC 



C 



7 = 1. 



= 1. 



But triangles having the same vertex and bases on the same line have 

 line areas proportional to their bases. Hence 



A'C B'A C'B 



7=1, 



BA' CB' AC 



which on changing the sign of the denominators becomes 



A'C B'A C'B 



AB' B'C C'A 



= - 1. 



The converse of this theorem is also easily proved. These two theorems 

 are the same for ordinary geometry as they are here, and in fact the 

 demonstrations here given are applicable to either geometry. 



In this geometry we have duals of the two preceding theorems, which 

 is not the case in ordinary geometry. 



