$A$ and $B$ are two points on the circumference of a circle center $O$. $C$ is a point on the major arc $AB$. Draw the lines $AC$, $BC$, $AO$, $BO$, and $CO$, extending the last line to a point $D$ inside the sector $AOB$. Prove that $\angle AOD$ is twice $\angle ACO$ and that $\angle BOD$ is twice angle $\angle BCO$. Hence show that the angle subtended by the minor arc $AB$ at the centre of the circle is twice the angle that it subtends at the circumference of the circle.