# 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 $180^{\circ}$.

###### Theorem 1.

[Euclid, Book III, Prop. 20] In any circle, a circumferential angle is half the size of the central angle subtending the same arc.

###### 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 $O$:

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

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

Similarly, $\triangle AOC$ is isosceles, and

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

and it follows that

 $\angle BOC=\angle FOB+\angle FOC=2\angle OAB+2\angle OAC=2\angle BAC$

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:

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

 $\angle COB=\angle COF-\angle BOF=2\angle OAC-2\angle OAB=2\angle BAC$

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. ∎

Title circumferential angle is half the corresponding central angle CircumferentialAngleIsHalfTheCorrespondingCentralAngle 2013-03-22 17:13:28 2013-03-22 17:13:28 rm50 (10146) rm50 (10146) 14 rm50 (10146) Theorem msc 51M04 AngleOfViewOfALineSegment RiemannSphere circumferential angle central angle