PHILLIPS AND MOORE. 



ALGEBRA OF PROJECTIVE GEOMETRY. 



781 



Since angles are proportional to the point vectors determined at a 

 fixed point by their intersections with the line f, we may consider them 

 as represented by vectors on f measured relative to F as infinite point. 

 Thus 



I = 1^ - - (RQIPF) = - (FPIQR). 



Consequently 



and 



(25) 



These equations, together with the two already found, form a set of 

 three independent relations connecting the six parts of a triangle. A 



Figure 28. 



triangle can be constructed having a given side and two given angles. 

 There cannot then be more than three relations connecting the six 

 parts, and therefore any other is a consequence of these. 



38. Point-line Invariant. If we have a fixed point A and a fixed 

 line LM and take a triangle with vertex A and base BC on the line LM, 

 we have from the above relation 



8 = bC=cB, (26) 



which shows that the quantity S is an invariant here just as in Case 1. 



