774 PROCEEDINGS OF THE AMERICAN ACADEMY. 



line areas proportional to the lengths of their bases, this relation can 

 be written 



A'C B'A B'C A'C BC;__ 



C'B' CB' C'A' A'B C'A 

 which is equivalent to 



C'A A'B BC _ 



C'B A'C B'A 



The converse of this theorem is immediately seen to be true. 



/4 



Figure 23. 



1/ through any point in the plane of a triangle ABC lines are 

 drawn to the vertices cutting the sides opposite A, B, C in A', B', C re- 

 spectively, then 



rC C^ FA __ 



A'B C'A B'C 

 For we have the identity (Figures 23 and 24), 



oh pc of oa pc ob _ 

 ocof'oa'pc'pbpa'" * 



which becomes, on rearranging the terms, 



OT-PC PF-PA 0^-OB ^^ 

 OA-OC^'oT'CBOC-PF • 



