‘ ae ; 
286 On Analytical Geometry. 
angles, and consequently independent of a, y and z. Let us make 
wmy=Ac=1,then , W422? 
A?= “= 
. : B2 75> 9” 1 
It thus appears that A and B may be treated as functions solely 0 
the angle BAC. Accenting, to designate the other angles, and we 
: have — zt =a? +y?4+2(1—2A)azy (1) | 
2? =x? +y?—2(1—2B)ay 
y? =x? +2? +2(1—2A’)az (2) 
; y? =x" +27 — 221 mepome ¢ 
% gay? +22 +2(1 — 2A” \yz (3) 
x? =y? +23 — 2(1 — 2B” yz 
From the identity of the second members of (1), (2) and (3), re 
sult . A +B=1 : NS ai 
: ; A” +Bv=—1 pes 
Adding the corresponding equations of (1) and (2), and dividing by ce 
2x, of (1) and (3) and dividing by 2y, and of (2) and (3) dividing 
by 2z, we get a 
x+(1—2A)y+-(1-—2A/)z= (4) 
(1—2A)ety+(1—2A")z=0 (5) 
(1-—2A’)~+(1—2A”)jy+z=0. (6) 
From these three equations, we easily derive the equation of condi- 
tion (1—2A)?-+(1 2A’)? 4. (1 2A”)? 4.1 —2(1 —2A) (1-24) 
.  (1-2A”)=0, between the angles of the triangle ABC. 
_ Resuming the values of A and B, (observing that the difference of 
Boe a the squares of two numbers is equal to the product of their sum 47 
_.. difference) and. 
(w+-y-++2) (e+y—z) (w—y+2) (—2+9+*) 
x . See i: y 
in the same manner 
16A‘B/= 2aury? “+ 2x? z? + Qy224 —(x*+y! +z") 
272 
rpg LEY? +2022 2y%2? = (wt yt +2") 
10M gopher Ser Sythe 
me AB 2 - AB 22 AB 
and consequently, AB y?” A’B 2? and AR 
% 
hat 
