54. If from an external point to a circle one secant is drawn through the center, and another so that the external segment equals the radius of the circle, then the angle between the secants will be measured by one third of the larger of the arcs intercepted between its sides.

From the treatise "On the Sphere and Cylinder".
54. $\angle B O A=\angle B A O=\alpha$, if $A B=r$. But $\angle C B O=2 \alpha$ as an external angle relative to triangle $A O B$. Triangle $B O C$ is isosceles, therefore, $\angle B C O=2 \alpha$. But $\angle C O D$ is an external angle relative to triangle $A O C$, therefore, $\angle C O D=\angle C A O+\angle A C O=\alpha+$
$+2 \alpha=3 \alpha$.