PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 755 



from F. The ratio of two vectors on a line is a difference of double 

 ratios involving the end points of the vectors and the point of inter- 

 section of their line with f. That ratio is then unchanged when we 

 project from F upon another line. Therefore distances as here con- 

 structed are proportional along any line to the corresponding vectors 

 and consequently satisfy our definition. 



In this case equal vectors are always of equal length, i. e. the oppo- 

 site sides of a parallelogram are equal (parallel lines intersecting on f ). 

 This is shown in Figure 8. Let ABCD be a parallelogram. Draw FA 

 and FB to cut CD in M and N. Then since the vectors MN and CD 

 are equal, 



AB = MN = CD. 



We shall see later that if F is not on f, equal vectors on different lines 

 are not in general of equal length. 



Figure 9. 



19. Point F not on line f. If F is not on f, the locus of points at a 

 given distance from a point A is a line through the point P in which 

 FA cuts f. The locus of points at a constant distance from P is the 

 line FP. The equation 



AB = ^ 



gives for fixed A a line b, the locus of points B, and b is the corre- 

 spondent of A in a certain correlation. If a certain point G and its 



