744 PROCEEDINGS OF THE AMERICAN ACADEMY. 



To prove the second relation connecting A, B, H we have from the 

 lemma 



(OQ|AAA) _ (BF|AH) 

 (OTIHK) ~(/.BSi\AK)' 



and, from the theory of cross ratios, 



(OQ|AAA)=^, 



(BP[AH) = ^, 



(/iBS|AAK) = 2. 



Therefore 



(OT|HK) = -^ or (OT|KH) = ^ + '* 



A+M V I / 2 



This last relation is equivalent to the statement that 



But 



K = i (aA + fxB). 

 Therefore 



XA + mB = (A + m) H. (2) 



8. From the construction of A + B the sum is seen at once to obey 

 the following algebraic laws : 



A + B = B + A, 



AA + AiA = (A + )u) A. 

 If A + B = A + C then B = C. 



The associative law 



(A + B) + C=A+(B + C) = A + B4-C 



evidently holds for points on a line passing through 0. To prove that 

 it holds for any three points A, B, C, project them on a line drawn 

 through from two different points P, Q on f Let the projections be 

 A', B', C and A", B", C". The sum A + B + C will project into 

 A' + B' + C and A" + B" -f C". The sum A + B + C will then be 



