50 UNIVERSITY OF VIRGINIA PUBLICATIONS 



5. If the triangle A B G he lettered in the positive direction so that 

 the area bounded is kept on the left in going around it, then the signs 

 of the angles APB, BPC,GPA subtended by the sides take care of 

 themselves, counter clockwise being the positive direction. Let x, y, z 

 be the coordinates (areal) of any point P, whose radial coordinates are 

 i\, r.-,, 1\, then 



sinAPB sinBPC sinCPA 2A 



z r^ X r-^ y r.-, '>'i'>\i's (10) 



The angles have the same signs as those of tlie corresponding coordinates 

 of P. 



Equations (10) show that the segments x i\, y r^, zi\ laid off at P in 

 the directions P A, P B, P G, each toward the vertex if positive and away 

 from it if negative, are in equilibrium, and their absolute values construct 

 a triangle similar to A'B'C. Now if P be the critical point satisfying the 

 necessary conditions of the problem it follows that 



xr, yr.2 zi\ (11) 



Since the directions of the segments xr^, yi\, zr. and a, /3, y equilibrated 

 at P must be the same, the signs of the coordinates of P must be the 

 same as the signs of the corresponding coordinates of Q. Multiply each 

 term of the first ratio in (11) by rj, the second by r^, the third by r^; then 

 multiply in the same way by a, p, y respectively. An easy composition 

 of the resulting ratios gives each member of (11) equal to 



in virtue of the identical relation 



X r,-+ y U-+ z r^-= x y c-+.v z h-+ y z a\ (13) 



which exists between the radial and areal coordinates of any point. Hence 

 from (12) the value of s at the critical point P is given by 



, 'S.xyc-. %xyc'- 



xyz 



(14) 



The right member of (13), 'S.xyc-, we represent by o-; the power of P 

 with respect to the circumcircle oi A B C is — a. We shall represent the 

 similar function %x y c''^ by o-'. 



