KöMaL 2021 C1690

The centre of a semicircle of unit radius and diameter $AB$ is $O$. The semicircle of diameter $OB$, centred at $K$ is drawn inside the larger semicircle. A ray drawn from point $A$ touches the small semicircle at point $C$. The perpendicular dropped from point $O$ to $AC$ intersects the arc of diameter $AB$ at a point $D$. Prove that the midpoint of line segment $BD$ is $C$.