circumferential angle is half the corresponding central angle
Consider a circle with center and two distinct points on the circle and . If is a third point on the circle not equal to either or , then the circumferential angle at subtending the arc is the angle . Here, by arc , we mean the arc of the circle that does not contain the points .
Similarly, the central angle subtending arc is the angle . The central angle corresponds to the arc measured on the same side of the circle as the angle itself. Note that if is a diameter of the circle, then the central angle is .
[Euclid, Book III, Prop. 20] In any circle, a circumferential angle is half the size of the central angle subtending the same arc.
There are actually several distinct cases. Consider in a circle with center , and draw as well as the chord containing both and :
Similarly, is isosceles, and
and it follows that
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 and are isosceles, so that again
Subtracting, we get
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|
|Date of creation||2013-03-22 17:13:28|
|Last modified on||2013-03-22 17:13:28|
|Last modified by||rm50 (10146)|