MAXIMUM -VXD MtXIMUM VALUES 0I-' A LINEAR FUXCTION. ETC. 49 



^ = ah + l3h+yk = 0. 1 



■ -|^ = am, + pm- + y)-(, = 0. J 

 Transposing the last terms to the right, squaring and adding these results 



and in like manner similar values for the cosines of the angles (r^r^), 

 {i\i\). Eliminating in turn I.,, m^ then Z3, nu there res\ilt 



sin(r2r, ) sin(r3;-i) sin (r^r,) (7) 



Also directly from (6) 



sin= ( r,u ) = ■!„. ( -ia'p-' + WY + Yo" — a'-l3'~ y' ) . 



-La'p' 



4^(a + ^ + y) ia + P — y)(a — l3 + y) (— '^ + ^ + y). (8) 



We tind like expressions for the sines of the other two angles. 



These conditions show that it is necessary that the absolute values 

 a', v. c' of a, p, y must be the sides of a triangle whose sides are parallel 

 to PA, P B, PC. Also (5) shows that segments a, /3, y laid off from P 

 along P A, P-B, PC are in equilibrium, each being directed toward a 

 vertex when positive and away from it when negative. 



The equation in areal coordinates 



(x + y + z) {a- + y — z) (x — y + z) {—x + y + z)=0, (9) 



represents four straight lines, the line at infinity and tln-ee straight lines 

 passing through the mid points D, E, F of the sides of the triangle ABC. 

 The expression on the left of (9) is positive when the point x, y, z is inside 

 D E F or in one of the vertical angles D, E, F ; it vanishes at any point 

 on the straight lines bounding this region, and is negative at all other 

 points in the plane. Therefore (8) shows that the point Q {a, /3, y) 

 must he m D E F OT one of its vertical angles. Hence a, (3, y must be all 

 positive or only one can be negative. 



The triangle A' B' C whose sides are a', V, c' , the absolute values of 

 a, /3, y, we shall call the reciprocal triangle of A B C. 



