1986 Putnam B2

Prove that there are only a finite number of possibilities for the

ordered triple $T=(x-y,y-z,z-x)$, where $x,y,z$ are complex numbers

satisfying the simultaneous equations

$$

x(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,

$$

and list all such triples $T$.

证明只有有限种可能性

有序三元组 $T=(x-y,y-z,z-x)$，其中 $x,y,z$ 是复数

满足联立方程组

$$

x(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,

$$

并列出所有这样的三元组$T$。