To find the center of a given circle or of a circle of which an arc is given

Given: Arc GH of a circle

Required: To find the center of the circle.

Procedure:

find the center of a circle

Select any three points on arc GH.

Label the points A, B, and C.

Use the segment command under the construct menu and construct segments AB and BC.

Construct the perpendicular bisectors of these two segments. This can be accomplished by following the basic perpendicular bisector construction; (help, midpoint) (help,

) or by using the tools in Sketchpad which follows:

Select segment AB.

Use the midpoint command under the construct menu to place a midpoint on AB. Label it D.

Select the midpoint D and segment AB, use the perpendicular line command under the construct menu and construct a perpendicular at D.

Select segment BC and repeat the above steps, but label the midpoint E.

Label the point of intersection of the two perpendicular bisectors point F.

F is the center of the given circle/arc.

Proof:

F is the intersection of the

bisectors of the sides of the triangle created by segments AB, BC, and CA.

Segment FA = FB = FC ( The

bisector of the sides of a triangle are concurrent in a point equidistant from the vertices.)

F is the center of the given circle/arc.

Sketch: Reposition points A, B, or C on the arc of the circle in the sketch below. Does the center remain the same?

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find center of circle

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