6. Given $\overrightarrow{O A}$ and $\overrightarrow{O B}$ are non-zero and non-collinear vectors. Let $\overrightarrow{O C}=\frac{1}{1+r} \overrightarrow{O A}+\frac{r}{1+r} \overrightarrow{O B}$. Define the set of points
$$
M=\left\{K \left\lvert\, \frac{\mid \overrightarrow{K A} \cdot \overrightarrow{K C}}{|\overrightarrow{K A}|}=\frac{\overrightarrow{K B} \cdot \overrightarrow{K C}}{|\overrightarrow{K B}|}\right.\right\} .
$$

When $K_{1} 、 K_{2} \in M$, if for any $r \geqslant 2$, the inequality $\left|\overrightarrow{K_{1} K_{2}}\right| \leqslant c|\overrightarrow{A B}|$ always holds, then the minimum value of the real number $c$ is $\qquad$