ma:s;imum and minimum values of a linear function, etc. 55 



which it attains A. We shall presently show that s is a minimum under the 

 circumstances, but first seek the position or positions of Q when F is at A. 

 Since cos A ^ — cos A' we have 



P' + r-'^' =-cosA. 



2/3y 

 The locus in areal coordinates 



if -{-z^~x- + 2yz cos A = (29) 



or as it can be written 



y -\- z — X — 2yz (1 — cos 4 ) = 



is a hyperbola, one branch of which passes through the points E and F. 

 It does not meet C B. The coordinates of its center, which is on the 

 median through A, are given by 



X 



'= y = z, 



1 + cos 4 



the actual value of x being — eot^^A. To construct the center, draw 

 the diameter of the circumcircle oi A B C through the mid point of side a ; 

 Join A to J the end of the diameter on the same side of a a.s A; tlxrough 

 the other end of the diameter draw a parallel to 4 J cutting the median 

 in the center of the hyperbola. The lines E D and F D, whose equations 

 are 



y -\- X — 2 = 0, 



X -\- z — y ^ 0, 



meet the hjrperbola each in only one finite point, E and F respectively ; the 

 asymptotes are therefore parallel to these lines and pass through the 

 previously constructed center. Hence the branch of the hyperbola (29) 

 passing through E and F lies wholly within the region for which 8^ is 

 positive, and therefore that arc of it interior to D E F contains all the 

 points a, /8, y at which A' is the supplement of A. In like manner the 

 arcs of the hyperbolae 



x^ -\- z- — y- -\- 2 X z cos 5 = 0, | 

 a-= -j- ,1/2 _ £2 _)_ 2 X 2/ cos C = 0, [ 



interior to D E F contain all points a, jS, y for which B' -\- B = -n- and 

 C" + C = TT respectively. 



