Given the circle, midpoint $O(0,0)$ and radius $r$, point $A(r\cos \beta,r\sin \beta)$, the angle $\beta$ is varying. Given the circle, midpoint $C(-2r,0)$ and radius $r$, point $B(-2r+r\cos(\alpha+\beta),r\sin(\alpha+\beta))$, the angle $\alpha$ is constant. The perpendicular bisector of $AB$ passes through the fixed point $D(-r,-r\cot \frac{\alpha}{2})$.