By eliminating a,b,c from the homogenous equations x=ab−c,y=bc−a,z=ca−b where a,b,c not all zero
xy+yz+zx=1
xy+yz+zx=−1
x+y+z=0
x+y+z=1
x=ab−c⇒a=xb−xc⇒a−xb+xc=0→(1)y=bc−a⇒b=yc−ya⇒ya+b−yc=0→(2)z=ca−b⇒c=za−zb⇒za−zb−c=0→(3)
By eliminating a.b,c from (1), (2), (3), we get 1−xxy1−yz−z−1=0
⇒1(−1−yz)+x(−y+yz)+x(−yz−z)=0⇒−1−yz−xy+xyz−xyz−zx=0⇒xy+yz+zx=−1