786 



Let 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



AD = A/ic, AE = \vb. 



Then by definition AB + AC = 2h, where H is the harmonic of f with 

 respect to D, E and h joins A to H. It was shown that the line AH 



divides the segment BC in the ratio and therefore the Hne must 



pass through B + C. It is evident that the magnitude of A (B + C) 



is equal to the magnitude of AB + AC, each being equal to A (yu + v), 

 and since these lines have two points in common they are then equal. 

 The distributive law for points and lines 



a (C + D) = aC + aD 



(38) 



can be proved as follows : Let a = AB where A and B are unit points. 

 Then the left member can be written 



AB (C + D), 



or, since the products are associative, 



A (BC + BD). 

 Let (Figure 30) H = i(C + D). 



Then BH = ^B (C + D) = i(BC + BD) 



