PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 765 



We thus have two means of finding the sum 



AA + fxB. 



The one is an anharmonic construction involving the points A,B and 

 the line f. The other is a harmonic or vector construction involving 

 the sectors A.A, nB and the point F. These constructions are both used 

 in ordinary geometry, but the one only for adding points, the other 

 only for adding lines. We have used both constructions in dual forms 

 for both purposes. 



>r 



Figure 17. 



§ 3. The Triangle. 



28. In this section we shall apply the notions of distance and angle, 

 as given in the previous section, to the study of plane figures, in par- 

 ticular to triangles. We have already seen that many properties are 

 quite different according as the point F is on the line f or not. We 

 shall therefore consider these two cases separately, taking first the 

 case in which F is not on f and then the other as a special case. 



Case I. Point F not on line f. 



29. Relation between distance and angle. A comparison of the 

 construction for equal distances with that for equal angles shows that 

 the two are given by the same diagram. The group of collineations 

 that leaves distance unaltered will then leave angle unaltered and con- 

 versely. We should therefore expect these quantities to be related. 



When A is fixed and 



AB = const. 



