PHILLIPS AND MOORE. — ALGEBRA OF PROJECTIVE GEOMETRY. 743 



Figure 4. 



Let M be the intersection of AB and CD, and project all points on f 

 from and M. The two ratios above can then be written as follows : 



(OQIAC) _ (LQJPS) _ 



(OR I BD) ~ (LR I PS) ~ ^^^ ' ^^^ 



(HP|AB) _(TP[QR) _ . 



(KS I CD) (TS I QR) ~ ^ ' ^^^ ' 

 but (RQjPS) = (SP|QR), 



which proves the lemma. 



If we apply the above lemma to Figure 3, replacing C by XA, D by 

 fiB, and K by ^ (AA + /xB), it is at once seen that H divides AB in 



the cross ratio — - with respect to f For, in this case. 



(KS|CD)=-1, 



(OQIAC) = i, 



(OR|BD) = -- 

 From which it follows that 



(HP|AB) = -^. 



A 

 This shows that H is independent of the position of 0. 



