# angles subtending arcs at the circumference and centre

$A$ 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.

• WLOG, assume $D$ is on the circumference of the circle so that $CD$ is a diameter. – Eric Towers Apr 27 '14 at 19:44
• i don't know what formal steps to take here – user145948 Apr 27 '14 at 20:05
• Have you drawn the diagram? – Eric Towers Apr 27 '14 at 20:05
• yeah, perhaps im just tired. – user145948 Apr 27 '14 at 20:08
• Do you know "the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the circle" ? – Eric Towers Apr 27 '14 at 20:10