6. Given $A_{i}\left(t_{i}+\frac{1}{t_{i}}, t_{i}-\frac{1}{t_{i}}\right)(i=1,2,3$,
4) are four distinct points on a plane. If these four points are concyclic, then $t_{1} 、 t_{2} 、 t_{3} 、 t_{4}$ should satisfy $\qquad$ .
6. $t_{1} t_{2} t_{3} t_{4}=1$.

Let the equation of the circle be
$$
x^{2}+y^{2}+D x+E y+F=0 \text {. }
$$

If $P\left(t+\frac{1}{t}, t-\frac{1}{t}\right)$ lies on the circle, then
$$
\left(t+\frac{1}{t}\right)^{2}+\left(t-\frac{1}{t}\right)^{2}+D\left(t+\frac{1}{t}\right)+E\left(t-\frac{1}{t}\right)+P=0 \text {. }
$$

Simplifying, we get
$$
2 t^{4}+(D+E) t^{3}+F t^{2}+(D-E) t+2=0 \text {. }
$$

Thus, $t_{i}(i=1,2,3,4)$ are the four roots of equation (1).
By the relationship between roots and coefficients, we have
$$
t_{1} t_{2} t_{3} t_{4}=1 \text {. }
$$

Note: It can be proven that $t_{1} t_{2} t_{3} t_{4}=1$ is a necessary and sufficient condition for $A_{i}(i=1,2,3,4)$ to be concyclic.