construct the center of a given circle

[Euclid, Book III, Prop. 1] Find the center ( of a given circle.

Since, in Euclidean geometryMathworldPlanetmath, a circle has one center only, it suffices to construct a point that is a center of the given circle.

Draw any chord AB¯ in the circle, and construct the perpendicular bisectorMathworldPlanetmath of AB¯, intersecting AB¯ in C, and the circle in D,E.

Let O be the center of the circle; we will show that O is the midpointMathworldPlanetmathPlanetmathPlanetmath of DE¯. Note that in the diagram below, O is purposely drawn not to lie on DE¯; the proof shows that this position is impossible and that in fact O lies on DE¯. It then follows easily that in fact O is the midpoint of DE¯.


Since O is the center of the circle, it follows that OA=OB. Since DE¯ bisects AB¯, we see in addition that AC=BC. ACO and BCO share their third side, OC¯. So by SSS, ACOBCO, and thus, using CPCTC, ACOBCO. But ACO+BCO=180, so ACO and BCO are each right anglesMathworldPlanetmathPlanetmath. Thus O in fact lies on DE¯.

However, since O is the center of the circle, it must be equidistant from D and E, and thus O is the midpoint of DE¯.

Title construct the center of a given circle
Canonical name ConstructTheCenterOfAGivenCircle
Date of creation 2013-03-22 17:13:41
Last modified on 2013-03-22 17:13:41
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 9
Author rm50 (10146)
Entry type Derivation
Classification msc 51M15
Classification msc 51-00
Related topic CompassAndStraightedgeConstructionOfCenterOfGivenCircle