PlanetMath: circumferential angle is half the corresponding central angle

Consider a circle with center $O$ and two distinct points on the circle $A$ and $B$ . If $C$ is a third point on the circle not equal to either $A$ or $B$ , then the circumferential angle at $C$ subtending the arc $AB$ is the angle $ACB$ . Here, by arc $AB$ , we mean the arc of the circle that does not contain the points $C$ .

Similarly, the central angle subtending arc $AB$ is the angle $AOB$ . The central angle corresponds to the arc $AB$ measured on the same side of the circle as the angle itself. Note that if $AB$ is a diameter of the circle, then the central angle is $180^{\circ}$ .

Proof. There are actually several distinct cases. Consider $\angle BAC$ in a circle with center $O$ , and draw $AO, BO, CO$ as well as the chord containing both $A$ and $O$ :

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In this case, the center of the circle lies between the arms of the circumferential angle. Now, since , $\triangle AOB$ is isosceles, and $\angle FOB$ is an exterior angle. Thus

$\displaystyle \angle FOB=\angle OAB + \angle OBA = 2\angle OA$

Similarly, $\triangle AOC$ is isosceles, and

$\displaystyle \angle FOC=\angle OAC + \angle OCA = 2\angle OA$

and it follows that

$\displaystyle \angle BOC=\angle FOB + \angle FOC = 2\angle OAB + 2\angle OAC = 2\angle BA$

proving the result.

A second case is the case in which both arms of the angle lie to one side of the circle's center:

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The proof is similar to the previous case, except that the angle in question is the difference rather than the sum of two known angles. Here we see that both $\triangle AOB$ and $\triangle AOC$ are isosceles, so that again

$\displaystyle \angle COF$	$\displaystyle = 2\angle OAC$
$\displaystyle \angle BOF$	$\displaystyle = 2\angle OAB$

Subtracting, we get$$\angle COB = \angle COF - \angle BOF = 2\angle OAC -2\angle OAB = 2\angle BA$$ as desired.

The final case is the case in which one arm of the angle goes through the center of the circle. This is a degenerate form of the first case, and the same proof follows through except that one of the angles is zero. $\qedsymbol$