construct the center of a given circle
[Euclid, Book III, Prop. 1] Find the center (http://planetmath.org/Center8) of a given circle.
Draw any chord in the circle, and construct the perpendicular bisector of , intersecting in , and the circle in .
Let be the center of the circle; we will show that is the midpoint of . Note that in the diagram below, is purposely drawn not to lie on ; the proof shows that this position is impossible and that in fact lies on . It then follows easily that in fact is the midpoint of .
Since is the center of the circle, it follows that . Since bisects , we see in addition that . and share their third side, . So by SSS, , and thus, using CPCTC, . But , so and are each right angles. Thus in fact lies on .
However, since is the center of the circle, it must be equidistant from and , and thus is the midpoint of .
|Title||construct the center of a given circle|
|Date of creation||2013-03-22 17:13:41|
|Last modified on||2013-03-22 17:13:41|
|Last modified by||rm50 (10146)|